Misplaced Pages

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75-400: Area is the measure of a region 's size on a surface . The area of a plane region or plane area refers to the area of a shape or planar lamina , while surface area refers to the area of an open surface or the boundary of a three-dimensional object . Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or

150-451: A σ {\displaystyle \sigma } -algebra over X . {\displaystyle X.} A set function μ {\displaystyle \mu } from Σ {\displaystyle \Sigma } to the extended real number line is called a measure if the following conditions hold: If at least one set E {\displaystyle E} has finite measure, then

225-399: A definite integral : The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes x and y the formula is: Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces ). For example, if the side surface of a cylinder (or any prism )

300-429: A greatest element μ sf . {\displaystyle \mu _{\text{sf}}.} We say the semifinite part of μ {\displaystyle \mu } to mean the semifinite measure μ sf {\displaystyle \mu _{\text{sf}}} defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for

375-514: A least measure μ 0 − ∞ . {\displaystyle \mu _{0-\infty }.} Also, we have μ = μ sf + μ 0 − ∞ . {\displaystyle \mu =\mu _{\text{sf}}+\mu _{0-\infty }.} We say the 0 − ∞ {\displaystyle \mathbf {0-\infty } } part of μ {\displaystyle \mu } to mean

450-409: A corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m), square centimetres (cm), square millimetres (mm), square kilometres (km), square feet (ft), square yards (yd), square miles (mi), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area

525-548: A countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure. For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [ k , k + 1 ] {\displaystyle [k,k+1]} for all integers k ; {\displaystyle k;} there are countably many such intervals, each has measure 1, and their union

600-485: A measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as Banach limits , the dual of L ∞ {\displaystyle L^{\infty }} and the Stone–Čech compactification . All these are linked in one way or another to

675-655: A measure on A . {\displaystyle {\cal {A}}.} We say μ {\displaystyle \mu } is semifinite to mean that for all A ∈ μ pre { + ∞ } , {\displaystyle A\in \mu ^{\text{pre}}\{+\infty \},} P ( A ) ∩ μ pre ( R > 0 ) ≠ ∅ . {\displaystyle {\cal {P}}(A)\cap \mu ^{\text{pre}}(\mathbb {R} _{>0})\neq \emptyset .} Semifinite measures generalize sigma-finite measures, in such

750-1556: A measure. If E 1 {\displaystyle E_{1}} and E 2 {\displaystyle E_{2}} are measurable sets with E 1 ⊆ E 2 {\displaystyle E_{1}\subseteq E_{2}} then μ ( E 1 ) ≤ μ ( E 2 ) . {\displaystyle \mu (E_{1})\leq \mu (E_{2}).} For any countable sequence E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } of (not necessarily disjoint) measurable sets E n {\displaystyle E_{n}} in Σ : {\displaystyle \Sigma :} μ ( ⋃ i = 1 ∞ E i ) ≤ ∑ i = 1 ∞ μ ( E i ) . {\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).} If E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are increasing (meaning that E 1 ⊆ E 2 ⊆ E 3 ⊆ … {\displaystyle E_{1}\subseteq E_{2}\subseteq E_{3}\subseteq \ldots } ) then

825-1712: A monotonically non-decreasing sequence converging to t . {\displaystyle t.} The monotonically non-increasing sequences { x ∈ X : f ( x ) > t n } {\displaystyle \{x\in X:f(x)>;t_{n}\}} of members of Σ {\displaystyle \Sigma } has at least one finitely μ {\displaystyle \mu } -measurable component, and { x ∈ X : f ( x ) ≥ t } = ⋂ n { x ∈ X : f ( x ) > t n } . {\displaystyle \{x\in X:f(x)\geq t\}=\bigcap _{n}\{x\in X:f(x)>t_{n}\}.} Continuity from above guarantees that μ { x ∈ X : f ( x ) ≥ t } = lim t n ↑ t μ { x ∈ X : f ( x ) > t n } . {\displaystyle \mu \{x\in X:f(x)\geq t\}=\lim _{t_{n}\uparrow t}\mu \{x\in X:f(x)>t_{n}\}.} The right-hand side lim t n ↑ t F ( t n ) {\displaystyle \lim _{t_{n}\uparrow t}F\left(t_{n}\right)} then equals F ( t ) = μ { x ∈ X : f ( x ) > t } {\displaystyle F(t)=\mu \{x\in X:f(x)>t\}} if t {\displaystyle t}

SECTION 10

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900-941: A null set. One defines μ ( Y ) {\displaystyle \mu (Y)} to equal μ ( X ) . {\displaystyle \mu (X).} If f : X → [ 0 , + ∞ ] {\displaystyle f:X\to [0,+\infty ]} is ( Σ , B ( [ 0 , + ∞ ] ) ) {\displaystyle (\Sigma ,{\cal {B}}([0,+\infty ]))} -measurable, then μ { x ∈ X : f ( x ) ≥ t } = μ { x ∈ X : f ( x ) > t } {\displaystyle \mu \{x\in X:f(x)\geq t\}=\mu \{x\in X:f(x)>t\}} for almost all t ∈ [ − ∞ , ∞ ] . {\displaystyle t\in [-\infty ,\infty ].} This property

975-425: A rectangle with length l and width w , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom . On

1050-948: A sense, semifinite once its 0 − ∞ {\displaystyle 0-\infty } part (the wild part) is taken away. Theorem (Luther decomposition)  —  For any measure μ {\displaystyle \mu } on A , {\displaystyle {\cal {A}},} there exists a 0 − ∞ {\displaystyle 0-\infty } measure ξ {\displaystyle \xi } on A {\displaystyle {\cal {A}}} such that μ = ν + ξ {\displaystyle \mu =\nu +\xi } for some semifinite measure ν {\displaystyle \nu } on A . {\displaystyle {\cal {A}}.} In fact, among such measures ξ , {\displaystyle \xi ,} there exists

1125-483: A shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics , the unit square is defined to have area one, and

1200-449: A special case of semifinite measures and a generalization of sigma-finite measures. Let X {\displaystyle X} be a set, let A {\displaystyle {\cal {A}}} be a sigma-algebra on X , {\displaystyle X,} and let μ {\displaystyle \mu } be a measure on A . {\displaystyle {\cal {A}}.} A measure

1275-462: A sphere was first obtained by Archimedes in his work On the Sphere and Cylinder . The formula is: where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus . Measure (mathematics) The intuition behind this concept dates back to ancient Greece , when Archimedes tried to calculate

1350-437: A way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.) The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to μ . {\displaystyle \mu .} It can be shown there

1425-438: Is a greatest measure with these two properties: Theorem (semifinite part)  —  For any measure μ {\displaystyle \mu } on A , {\displaystyle {\cal {A}},} there exists, among semifinite measures on A {\displaystyle {\cal {A}}} that are less than or equal to μ , {\displaystyle \mu ,}

1500-488: Is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory ) are Radon measures . Radon measures have an alternative definition in terms of linear functionals on the locally convex topological vector space of continuous functions with compact support . This approach

1575-1052: Is a point of continuity of F . {\displaystyle F.} Since F {\displaystyle F} is continuous almost everywhere, this completes the proof. Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I {\displaystyle I} and any set of nonnegative r i , i ∈ I {\displaystyle r_{i},i\in I} define: ∑ i ∈ I r i = sup { ∑ i ∈ J r i : | J | < ∞ , J ⊆ I } . {\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<;\infty ,J\subseteq I\right\rbrace .} That is, we define

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1650-466: Is a unique t 0 ∈ { − ∞ } ∪ [ 0 , + ∞ ) {\displaystyle t_{0}\in \{-\infty \}\cup [0,+\infty )} such that F {\displaystyle F} is infinite to the left of t {\displaystyle t} (which can only happen when t 0 ≥ 0 {\displaystyle t_{0}\geq 0} ) and finite to

1725-471: Is an imperial and US unit of measure for area . One square mile is equal to the area of a square with each side measuring a length of one mile . One square mile is equal to: One square mile is also equivalent to: Square miles should not be confused with miles square, a square region with each side having a length of the value given. For example, a region which is 20 miles square ( 20 miles × 20 miles ) has an area of 400 sq mi ;

1800-445: Is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r , and the width is half the circumference of the circle, or π r . Thus, the total area of the circle is π r : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of

1875-474: Is called a measurable space , and the members of Σ {\displaystyle \Sigma } are called measurable sets . A triple ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is called a measure space . A probability measure is a measure with total measure one – that is, μ ( X ) = 1. {\displaystyle \mu (X)=1.} A probability space

1950-418: Is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone , the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature , it cannot be flattened out. The formula for the surface area of

2025-519: Is equivalent to the statement that the ideal of null sets is κ {\displaystyle \kappa } -complete. A measure space ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is called finite if μ ( X ) {\displaystyle \mu (X)} is a finite real number (rather than ∞ {\displaystyle \infty } ). Nonzero finite measures are analogous to probability measures in

2100-465: Is false without the assumption that at least one of the E n {\displaystyle E_{n}} has finite measure. For instance, for each n ∈ N , {\displaystyle n\in \mathbb {N} ,} let E n = [ n , ∞ ) ⊆ R , {\displaystyle E_{n}=[n,\infty )\subseteq \mathbb {R} ,} which all have infinite Lebesgue measure, but

2175-460: Is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

2250-414: Is measurable. A measure can be extended to a complete one by considering the σ-algebra of subsets Y {\displaystyle Y} which differ by a negligible set from a measurable set X , {\displaystyle X,} that is, such that the symmetric difference of X {\displaystyle X} and Y {\displaystyle Y} is contained in

2325-532: Is necessarily of finite variation , hence complex measures include finite signed measures but not, for example, the Lebesgue measure . Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure ; these are used in functional analysis for the spectral theorem . When it

Area - Misplaced Pages Continue

2400-406: Is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination , while signed measures are the linear closure of positive measures. Another generalization is the finitely additive measure , also known as a content . This is the same as

2475-427: Is related to the definition of determinants in linear algebra , and is a basic property of surfaces in differential geometry . In analysis , the area of a subset of the plane is defined using Lebesgue measure , though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through

2550-466: Is said to be s-finite if it is a countable sum of finite measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes . If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable ; examples of such sets include the Vitali set , and the non-measurable sets postulated by

2625-1000: Is semifinite then μ = μ sf . {\displaystyle \mu =\mu _{\text{sf}}.} Every 0 − ∞ {\displaystyle 0-\infty } measure that is not the zero measure is not semifinite. (Here, we say 0 − ∞ {\displaystyle 0-\infty } measure to mean a measure whose range lies in { 0 , + ∞ } {\displaystyle \{0,+\infty \}} : ( ∀ A ∈ A ) ( μ ( A ) ∈ { 0 , + ∞ } ) . {\displaystyle (\forall A\in {\cal {A}})(\mu (A)\in \{0,+\infty \}).} ) Below we give examples of 0 − ∞ {\displaystyle 0-\infty } measures that are not zero measures. Measures that are not semifinite are very wild when restricted to certain sets. Every measure is, in

2700-576: Is spatial distribution of mass (see for example, gravity potential ), or another non-negative extensive property , conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below. Measure theory is used in machine learning. One example is the Flow Induced Probability Measure in GFlowNet. Let μ {\displaystyle \mu } be

2775-820: Is such that μ { x ∈ X : f ( x ) > t } = + ∞ {\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } then monotonicity implies μ { x ∈ X : f ( x ) ≥ t } = + ∞ , {\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty ,} so that F ( t ) = G ( t ) , {\displaystyle F(t)=G(t),} as required. If μ { x ∈ X : f ( x ) > t } = + ∞ {\displaystyle \mu \{x\in X:f(x)>t\}=+\infty } for all t {\displaystyle t} then we are done, so assume otherwise. Then there

2850-399: Is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures . Some important measures are listed here. Other 'named' measures used in various theories include: Borel measure , Jordan measure , ergodic measure , Gaussian measure , Baire measure , Radon measure , Young measure , and Loeb measure . In physics an example of a measure

2925-471: Is the entire real line. Alternatively, consider the real numbers with the counting measure , which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to

3000-467: Is the square metre, which is considered an SI derived unit . Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m. This is equivalent to 6 million square millimetres. Other useful conversions are: In non-metric units,

3075-644: Is used in connection with Lebesgue integral . Both F ( t ) := μ { x ∈ X : f ( x ) > t } {\displaystyle F(t):=\mu \{x\in X:f(x)>t\}} and G ( t ) := μ { x ∈ X : f ( x ) ≥ t } {\displaystyle G(t):=\mu \{x\in X:f(x)\geq t\}} are monotonically non-increasing functions of t , {\displaystyle t,} so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative to

Area - Misplaced Pages Continue

3150-459: The Cartesian coordinates ( x i , y i ) {\displaystyle (x_{i},y_{i})} ( i =0, 1, ..., n -1) of whose n vertices are known, the area is given by the surveyor's formula : where when i = n -1, then i +1 is expressed as modulus n and so refers to 0. The most basic area formula is the formula for the area of a rectangle . Given

3225-549: The Hausdorff paradox and the Banach–Tarski paradox . For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure , while such a function with values in the complex numbers is called a complex measure . Observe, however, that complex measure

3300-492: The Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'. Let X {\displaystyle X} be a set, let A {\displaystyle {\cal {A}}} be a sigma-algebra on X , {\displaystyle X,} and let μ {\displaystyle \mu } be

3375-512: The area of a circle . But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel , Henri Lebesgue , Nikolai Luzin , Johann Radon , Constantin Carathéodory , and Maurice Fréchet , among others. Let X {\displaystyle X} be a set and Σ {\displaystyle \Sigma }

3450-479: The axiom of choice . Contents remain useful in certain technical problems in geometric measure theory ; this is the theory of Banach measures . A charge is a generalization in both directions: it is a finitely additive, signed measure. (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range its a bounded subset of R .) Square mile The square mile (abbreviated as sq mi and sometimes as mi )

3525-475: The hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad , the hectad , and the myriad . The acre is also commonly used to measure land areas, where An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns , such that: The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics . In South Asia (mainly Indians), although

3600-675: The intersection of the sets E n {\displaystyle E_{n}} is measurable; furthermore, if at least one of the E n {\displaystyle E_{n}} has finite measure then μ ( ⋂ i = 1 ∞ E i ) = lim i → ∞ μ ( E i ) = inf i ≥ 1 μ ( E i ) . {\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).} This property

3675-417: The surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. For a non-self-intersecting ( simple ) polygon,

3750-964: The union of the sets E n {\displaystyle E_{n}} is measurable and μ ( ⋃ i = 1 ∞ E i )   =   lim i → ∞ μ ( E i ) = sup i ≥ 1 μ ( E i ) . {\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)~=~\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).} If E 1 , E 2 , E 3 , … {\displaystyle E_{1},E_{2},E_{3},\ldots } are measurable sets that are decreasing (meaning that E 1 ⊇ E 2 ⊇ E 3 ⊇ … {\displaystyle E_{1}\supseteq E_{2}\supseteq E_{3}\supseteq \ldots } ) then

3825-554: The 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , but did not identify the constant of proportionality . Eudoxus of Cnidus , also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used

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3900-566: The Lebesgue measure. If t < 0 {\displaystyle t<0} then { x ∈ X : f ( x ) ≥ t } = X = { x ∈ X : f ( x ) > t } , {\displaystyle \{x\in X:f(x)\geq t\}=X=\{x\in X:f(x)>t\},} so that F ( t ) = G ( t ) , {\displaystyle F(t)=G(t),} as desired. If t {\displaystyle t}

3975-406: The amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle ); by synecdoche , "area" sometimes is used to refer to the region, as in a " polygonal area ". The area of

4050-470: The area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula , for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of

4125-505: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula , for

4200-423: The area of any other shape or surface is a dimensionless real number . There are several well-known formulas for the areas of simple shapes such as triangles , rectangles , and circles . Using these formulas, the area of any polygon can be found by dividing the polygon into triangles . For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining

4275-531: The area of plane figures was a major motivation for the historical development of calculus . For a solid shape such as a sphere , cone, or cylinder, the area of its boundary surface is called the surface area . Formulas for the surface areas of simple shapes were computed by the ancient Greeks , but computing the surface area of a more complicated shape usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area

4350-415: The areas of the approximate parallelograms is exactly π r , which is the area of the circle. This argument is actually a simple application of the ideas of calculus . In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus . Using modern methods, the area of a circle can be computed using

4425-406: The condition of non-negativity is dropped, and μ {\displaystyle \mu } takes on at most one of the values of ± ∞ , {\displaystyle \pm \infty ,} then μ {\displaystyle \mu } is called a signed measure . The pair ( X , Σ ) {\displaystyle (X,\Sigma )}

4500-416: The conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 12 = 12 × 12. Similarly: In addition, conversion factors include: There are several other common units for area. The are was the original unit of area in the metric system , with: Though the are has fallen out of use,

4575-519: The countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: In

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4650-637: The following hold: ⋃ α ∈ λ X α ∈ Σ {\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma } μ ( ⋃ α ∈ λ X α ) = ∑ α ∈ λ μ ( X α ) . {\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).} The second condition

4725-422: The intersection is empty. A measurable set X {\displaystyle X} is called a null set if μ ( X ) = 0. {\displaystyle \mu (X)=0.} A subset of a null set is called a negligible set . A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set

4800-405: The left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of

4875-864: The measure μ 0 − ∞ {\displaystyle \mu _{0-\infty }} defined in the above theorem. Here is an explicit formula for μ 0 − ∞ {\displaystyle \mu _{0-\infty }} : μ 0 − ∞ = ( sup { μ ( B ) − μ sf ( B ) : B ∈ P ( A ) ∩ μ sf pre ( R ≥ 0 ) } ) A ∈ A . {\displaystyle \mu _{0-\infty }=(\sup\{\mu (B)-\mu _{\text{sf}}(B):B\in {\cal {P}}(A)\cap \mu _{\text{sf}}^{\text{pre}}(\mathbb {R} _{\geq 0})\})_{A\in {\cal {A}}}.} Localizable measures are

4950-432: The other hand, if geometry is developed before arithmetic , this formula can be used to define multiplication of real numbers . Most other simple formulas for area follow from the method of dissection . This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle , as shown in figure to

5025-418: The parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons . The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk ) is based on a similar method. Given a circle of radius r , it is possible to partition the circle into sectors , as shown in the figure to the right. Each sector

5100-563: The requirement μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0} is met automatically due to countable additivity: μ ( E ) = μ ( E ∪ ∅ ) = μ ( E ) + μ ( ∅ ) , {\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),} and therefore μ ( ∅ ) = 0. {\displaystyle \mu (\varnothing )=0.} If

5175-872: The right. Arguing as above, μ { x ∈ X : f ( x ) ≥ t } = + ∞ {\displaystyle \mu \{x\in X:f(x)\geq t\}=+\infty } when t < t 0 . {\displaystyle t<t_{0}.} Similarly, if t 0 ≥ 0 {\displaystyle t_{0}\geq 0} and F ( t 0 ) = + ∞ {\displaystyle F\left(t_{0}\right)=+\infty } then F ( t 0 ) = G ( t 0 ) . {\displaystyle F\left(t_{0}\right)=G\left(t_{0}\right).} For t > t 0 , {\displaystyle t>t_{0},} let t n {\displaystyle t_{n}} be

5250-404: The semifinite part: Since μ sf {\displaystyle \mu _{\text{sf}}} is semifinite, it follows that if μ = μ sf {\displaystyle \mu =\mu _{\text{sf}}} then μ {\displaystyle \mu } is semifinite. It is also evident that if μ {\displaystyle \mu }

5325-421: The sense that any finite measure μ {\displaystyle \mu } is proportional to the probability measure 1 μ ( X ) μ . {\displaystyle {\frac {1}{\mu (X)}}\mu .} A measure μ {\displaystyle \mu } is called σ-finite if X {\displaystyle X} can be decomposed into

5400-606: The sum of the r i {\displaystyle r_{i}} to be the supremum of all the sums of finitely many of them. A measure μ {\displaystyle \mu } on Σ {\displaystyle \Sigma } is κ {\displaystyle \kappa } -additive if for any λ < κ {\displaystyle \lambda <\kappa } and any family of disjoint sets X α , α < λ {\displaystyle X_{\alpha },\alpha <\lambda }

5475-409: The tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle . (The circumference is 2 π r , and the area of a triangle is half the base times the height, yielding the area π r for the disk.) Archimedes approximated

5550-509: The use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. An approach to defining what is meant by "area" is through axioms . "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Every unit of length has

5625-419: The value of π (and hence the area of a unit-radius circle) with his doubling method , in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon , then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons ). Heron of Alexandria found what

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