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65-470: MCT may refer to: Astronomy [ edit ] A Maksutov–Cassegrain telescope Morning Civil Twilight , from when the center of the Sun is less than 6° below the horizon to sunrise. Biochemistry [ edit ] Medium-chain triglycerides , a class of fat with specific dietary and technical properties (also MCT oil) Monocarboxylate transporters ,

130-445: A N > c − ε {\displaystyle c\geq a_{N}>c-\varepsilon } , since otherwise c − ε {\displaystyle c-\varepsilon } is a strictly smaller upper bound of { a n } {\displaystyle \{a_{n}\}} , contradicting the definition of the supremum c {\displaystyle c} . Then since (

195-524: A i , k ≤ ∑ i sup k a i , k {\displaystyle \sum _{i}a_{i,k}\leq \sum _{i}\sup _{k}a_{i,k}} so sup k ∑ i a i , k ≤ ∑ i sup k a i , k {\displaystyle \sup _{k}\sum _{i}a_{i,k}\leq \sum _{i}\sup _{k}a_{i,k}} . Conversely, we can interchange sup and sum for finite sums by reverting to

260-403: A i , k ≤ a i , k + 1 {\displaystyle a_{i,k}\leq a_{i,k+1}} for all i , k {\displaystyle i,k} . Then Since a i , k ≤ sup k a i , k {\displaystyle a_{i,k}\leq \sup _{k}a_{i,k}} we have ∑ i

325-443: A i , k ≤ sup k ∑ i = 1 ∞ a i , k {\displaystyle \sum _{i=1}^{\infty }\sup _{k}a_{i,k}\leq \sup _{k}\sum _{i=1}^{\infty }a_{i,k}} . The theorem states that if you have an infinite matrix of non-negative real numbers a i , k ≥ 0 {\displaystyle a_{i,k}\geq 0} such that

390-503: A i , k ≤ ∑ K i {\displaystyle \sum _{i}a_{i,k}\leq \sum K_{i}} are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column" sup k a i , k {\displaystyle \sup _{k}a_{i,k}} which element wise is the supremum over the row. Consider the expansion Now set for i ≤ k {\displaystyle i\leq k} and

455-527: A i , k = 0 {\displaystyle a_{i,k}=0} for i > k {\displaystyle i>k} , then 0 ≤ a i , k ≤ a i , k + 1 {\displaystyle 0\leq a_{i,k}\leq a_{i,k+1}} with sup k a i , k = 1 i ! < ∞ {\displaystyle \sup _{k}a_{i,k}={\frac {1}{i!}}<\infty } and The right hand side

520-437: A n {\displaystyle \lim _{n\to \infty }a_{n}=c=\sup _{n}a_{n}} . The proof of the (B) part is analogous or follows from (A) by considering { − a n } n ∈ N {\displaystyle \{-a_{n}\}_{n\in \mathbb {N} }} . If ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }}

585-409: A n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} is non decreasing, and c {\displaystyle c} is an upper bound, for every n > N {\displaystyle n>N} , we have Hence, by definition lim n → ∞ a n = c = sup n

650-444: A n } n ∈ N {\displaystyle \{a_{n}\}_{n\in \mathbb {N} }} be the set of values of ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} . By assumption, { a n } {\displaystyle \{a_{n}\}} is non-empty and bounded above by K {\displaystyle K} . By

715-612: A jet engine rating Mercury cadmium telluride (HgCdTe), a semiconductor with a particularly narrow bandgap used in mid- and far-infrared detectors (e.g. night vision) MOS-controlled thyristor , a type of thyristor (a solid-state semiconductor device) Multi Cable Transits , A fire-stop is a fire protection system made of various components used to seal openings and joints in fire-resistance rated wall and/or floor assemblies. For penetrating cables, these can also be called as Multi Cable Transits (MCTs). Medicine [ edit ] Mast cell tumor Methacholine challenge test ,

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780-412: A Maksutov–Cassegrain. Commercial use of Gregory's design was explicitly reserved for Perkin–Elmer but was published as an amateur telescope design in a 1957 issue of Sky and Telescope in f /15 and f /23 variations. Most Maksutovs manufactured today are this type of 'Cassegrain' design (called either a "Gregory–Maksutov" or "Spot-Maksutov") that use all-spherical surfaces and have, as secondary,

845-1268: A countable disjoint union of measurable sets and likewise S k = ∐ 1 ≤ i ≤ k S i ∖ S i − 1 {\displaystyle S_{k}=\coprod _{1\leq i\leq k}S_{i}\setminus S_{i-1}} as a finite disjoint union. Therefore μ ( S k ) = ∑ i = 1 k μ ( S i ∖ S i − 1 ) {\displaystyle \mu (S_{k})=\sum _{i=1}^{k}\mu (S_{i}\setminus S_{i-1})} , and μ ( S ) = ∑ i = 1 ∞ μ ( S i ∖ S i − 1 ) {\displaystyle \mu (S)=\sum _{i=1}^{\infty }\mu (S_{i}\setminus S_{i-1})} so μ ( S ) = sup k μ ( S k ) {\displaystyle \mu (S)=\sup _{k}\mu (S_{k})} . Set f = sup k f k {\displaystyle f=\sup _{k}f_{k}} . Denote by SF ⁡ ( f ) {\displaystyle \operatorname {SF} (f)}

910-634: A degree of freedom in correcting the optical system by changing the radius of curvature of the secondary is lost, since that radius is the same as that of the rear meniscus face. Gregory himself, in a second, faster ( f /15 ) design, resorted to aspherization of the front corrector surface (or the primary mirror) in order to reduce aberrations. This has led to other designs with aspheric or additional elements to further reduce off-axis aberration. This type of Maksutov-Cassegrain's high focal ratio and narrower field of view makes them more suitable for lunar and planetary imaging and any other type of observing where

975-696: A family of proton-linked plasma membrane transporters that carry molecules having one carboxylate group (monocarboxylates), such as lactate and pyruvate, across biological membranes. Methyl halide transferase , an enzyme Companies [ edit ] Marine Current Turbines McClatchy-Tribune Information Services , the news service formerly known as Knight Ridder Tribune Computing [ edit ] Microsoft Certified Trainer Mobile Computer Terminal Multi Core Timer in Exynos (system on chip) processors Engineering [ edit ] Maximum continuous thrust , an aviation abbreviation for

1040-582: A finite limit if and only if the sequence is bounded . There is a variant of the proposition above where we allow unbounded sequences in the extended real numbers, the real numbers with ∞ {\displaystyle \infty } and − ∞ {\displaystyle -\infty } added. In the extended real numbers every set has a supremum (resp. infimum ) which of course may be ∞ {\displaystyle \infty } (resp. − ∞ {\displaystyle -\infty } ) if

1105-608: A measurable set. Let { f k } k = 1 ∞ {\displaystyle \{f_{k}\}_{k=1}^{\infty }} be a pointwise non-decreasing sequence of ( Σ , B R ¯ ≥ 0 ) {\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})} - measurable non-negative functions, i.e. each function f k : X → [ 0 , + ∞ ] {\displaystyle f_{k}:X\to [0,+\infty ]}

1170-413: A measurable space. Consider a simple ( Σ , B R ≥ 0 ) {\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})} -measurable non-negative function s : Ω → R ≥ 0 {\displaystyle s:\Omega \to {\mathbb {R} _{\geq 0}}} . For

1235-1049: A measurable subset S ∈ Σ {\displaystyle S\in \Sigma } , define Then ν s {\displaystyle \nu _{s}} is a measure on ( Ω , Σ ) {\displaystyle (\Omega ,\Sigma )} . Write s = ∑ k = 1 n c k ⋅ 1 A k , {\displaystyle s=\sum _{k=1}^{n}c_{k}\cdot {\mathbf {1} }_{A_{k}},} with c k ∈ R ≥ 0 {\displaystyle c_{k}\in {\mathbb {R} }_{\geq 0}} and measurable sets A k ∈ Σ {\displaystyle A_{k}\in \Sigma } . Then Since finite positive linear combinations of countably additive set functions are countably additive, to prove countable additivity of ν s {\displaystyle \nu _{s}} it suffices to prove that,

1300-618: A measure, and S = ⋃ i = 1 ∞ S i {\displaystyle S=\bigcup _{i=1}^{\infty }S_{i}} , where is a non-decreasing chain with all its sets μ {\displaystyle \mu } -measurable. Then Set S 0 = ∅ {\displaystyle S_{0}=\emptyset } , then we decompose S = ∐ 1 ≤ i S i ∖ S i − 1 {\displaystyle S=\coprod _{1\leq i}S_{i}\setminus S_{i-1}} as

1365-480: A medical test to assess the degree of a bronchial hyperresponsiveness (e.g. in asthma ). Munich Chronotype Questionnaire (MCTQ), a standardized test for evaluating patient sleep/wake patterns Ketogenic diet (MCT diet) Micro Computed Tomography Psychology [ edit ] Metacognitive therapy Transportation [ edit ] Mars Colonial Transporter , a proposed interplanetary transportation system by SpaceX Madison County Transit ,

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1430-540: A narrow field high power view is a plus, such as resolving tightly packed globular clusters and double stars . The most notable early amateur astronomical type was the Questar 3-1/2 Maksutov Cassegrain introduced in 1954, a small-run, expensive model still available on the consumer market. The mid-1970s saw the introduction of mass-produced models by some of the major commercial manufacturers. More recently, low-cost Russian and, lately, Chinese mass-production has pushed

1495-488: A non-decreasing bounded -above sequence of real numbers a 1 ≤ a 2 ≤ a 3 ≤ . . . ≤ K {\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq ...\leq K} converges to its smallest upper bound, its supremum . Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum . In particular, infinite sums of non-negative numbers converge to

1560-609: A public transportation service in Madison County, Illinois Muscat International Airport , the airport serving Muscat, Oman (IATA code is MCT) Other [ edit ] Minnesota Chippewa Tribe Mengisa language (ISO 639 code: mct) Marine Combat Training, a training course at the United States Marine Corps School of Infantry Monotone convergence theorem , in mathematics Motivation crowding theory , in economics Member of

1625-703: A slight generalization in 1906 of an earlier result by Henri Lebesgue . Let B R ¯ ≥ 0 {\displaystyle \operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}}} denotes the σ {\displaystyle \sigma } -algebra of Borel sets on the upper extended non negative real numbers [ 0 , + ∞ ] {\displaystyle [0,+\infty ]} . By definition, B R ¯ ≥ 0 {\displaystyle \operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}}} contains

1690-400: A small aluminized spot on the inner face of the corrector. This has the advantage of simplifying construction. It also has the advantage of fixing the alignment of the secondary and eliminates the need for a 'spider' that would cause diffraction spikes. The disadvantage is that, if all spherical surfaces are used, such systems have to have focal ratios above f /15 to avoid aberrations. Also,

1755-905: A train of refugees from Leningrad. Maksutov is described as patenting his design in May, August, or October 1941 and building a "Maksutov– Gregorian "-style prototype in October 1941. Maksutov came up with the unique idea using an "achromatic corrector", a corrector made of a single type of glass with a weak negative meniscus shape that departed from the pure concentric spherical symmetrical shape to correct chromatic aberration. Similar independent meniscus telescope designs were also patented in 1941: Albert Bouwers (his 1941 concentric meniscus telescope ), K. Penning and Dennis Gabor (a catadioptric non-monocentric design). Wartime secrecy kept these inventors from knowing about each other's designs, leading to each being an independent invention. Maksutov's 1944 design

1820-443: A wide field of view , with one-fourth the coma of a similar standard Newtonian and one-half the coma of a Schmidt-Newtonian . Diffraction can also be minimized by using a high focal ratio with a proportionally small diagonal mirror mounted on the corrector, allowing this design to achieve contrast and image quality approaching that of unobstructed high-end refractors (although with some vignetting when used photographically). Like

1885-464: Is ( Σ , B R ¯ ≥ 0 ) {\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})} -measurable and for every k ≥ 1 {\displaystyle {k\geq 1}} and every x ∈ X {\displaystyle {x\in X}} , Then

1950-471: Is a monotone sequence of real numbers , i.e., if a n ≤ a n + 1 {\displaystyle a_{n}\leq a_{n+1}} for every n ≥ 1 {\displaystyle n\geq 1} or a n ≥ a n + 1 {\displaystyle a_{n}\geq a_{n+1}} for every n ≥ 1 {\displaystyle n\geq 1} , then this sequence has

2015-430: Is a non decreasing sequence in k {\displaystyle k} , therefore The following result is a generalisation of the monotone convergence of non negative sums theorem above to the measure theoretic setting. It is a cornerstone of measure and integration theory with many applications and has Fatou's lemma and the dominated convergence theorem as direct consequence. It is due to Beppo Levi , who proved

MCT - Misplaced Pages Continue

2080-458: Is different from Wikidata All article disambiguation pages All disambiguation pages Maksutov%E2%80%93Cassegrain telescope Dmitri Maksutov may have been working with the idea of pairing a spherical primary mirror in conjunction with a negative meniscus lens as far back as 1936. His notes from that time on the function of Mangin mirrors , an early catadioptric spotlight reflector consisting of negative lens with silvering on

2145-463: Is enough that there is a null set N {\displaystyle N} such that the sequence { f n ( x ) } {\displaystyle \{f_{n}(x)\}} non-decreases for every x ∈ X ∖ N . {\displaystyle {x\in X\setminus N}.} To see why this is true, we start with an observation that allowing

2210-516: Is the indicator function of X i {\displaystyle X_{i}} , are easily seen to be measurable and f ⋅ 1 X 1 ≤ f ⋅ 1 X 2 {\displaystyle f\cdot {\mathbf {1} }_{X_{1}}\leq f\cdot {\mathbf {1} }_{X_{2}}} . Now apply 1 . Lemma 2. Let ( Ω , Σ , μ ) {\displaystyle (\Omega ,\Sigma ,\mu )} be

2275-591: The Association of Corporate Treasurers , a professional organization Miss Chinese Toronto Pageant , the annual Toronto beauty pageant for Chinese Canadians Manganese cyclopentadienyl tricarbonyl , an antiknock additive for gasoline Mickelson Clarified Translation , in Modern English Bible translations Multichannel television Mutable collagenous tissue See also [ edit ] MCTS (disambiguation) Topics referred to by

2340-582: The Schmidt camera . Like the Schmidt camera, the Maksutov camera has a curved focal plane. Monotone convergence theorem In the mathematical field of real analysis , the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences , i.e. sequences that are non- increasing , or non- decreasing . In its simplest form, it says that

2405-439: The least-upper-bound property of real numbers, c = sup n { a n } {\textstyle c=\sup _{n}\{a_{n}\}} exists and c ≤ K {\displaystyle c\leq K} . Now, for every ε > 0 {\displaystyle \varepsilon >0} , there exists N {\displaystyle N} such that c ≥

2470-558: The limit definition, so ∑ i = 1 N sup k a i , k = sup k ∑ i = 1 N a i , k ≤ sup k ∑ i = 1 ∞ a i , k {\displaystyle \sum _{i=1}^{N}\sup _{k}a_{i,k}=\sup _{k}\sum _{i=1}^{N}a_{i,k}\leq \sup _{k}\sum _{i=1}^{\infty }a_{i,k}} hence ∑ i = 1 ∞ sup k

2535-560: The Maksutov–Cassegrain, the overall diameter of the optical system is limited, due to the mass of the corrector plate. Synta Taiwan currently produces a 190 mm version under the Sky-Watcher brand as does Explore Scientific with a 152 mm version designed in collaboration with astronomer David Levy . The Maksutov system can be used in a (rare) type of prime-focus ultra-wide-field astronomical camera design similar to

2600-420: The assumptions of the theorem, Note that the second chain of equalities follows from monoticity of the integral (lemma 2 below). Thus we can also write the conclusion of the theorem as with the tacit understanding that the limits are allowed to be infinite. Remark 3. The theorem remains true if its assumptions hold μ {\displaystyle \mu } -almost everywhere. In other words, it

2665-479: The back side, include a sketch of a Mangin mirror with the mirror part and the negative lens separated into two elements. Maksutov seems to have picked up the idea again in 1941 as a variation on an earlier design that paired a spherical mirror with a negative lens, Bernhard Schmidt 's 1931 " Schmidt camera ". Maksutov claimed to have come up with the idea of replacing the complex Schmidt corrector plate with an all-spherical "meniscus corrector plate" while riding in

MCT - Misplaced Pages Continue

2730-505: The definition of the Lebesgue integral for a non-negative function). Remark 4. The proof below does not use any properties of the Lebesgue integral except those established here. The theorem, thus, can be used to prove other basic properties, such as linearity, pertaining to Lebesgue integration. This proof does not rely on Fatou's lemma ; however, we do explain how that lemma might be used. Those not interested in this independency of

2795-406: The focus of the primary mirror . Most types use full-aperture correctors and are therefore not very large, since the corrector plate rapidly becomes prohibitively large, heavy and expensive as the aperture increases, with very long cool-down times to reach optimal optical performance. Most commercial manufacturers usually stop at 180 mm (7 in). Maksutov's design notes from 1941 explored

2860-458: The inner surface of the meniscus corrector, sometimes similar to the corrector/mirror holder configurations found in commercial Schmidt–Cassegrains . This provides an extra degree of freedom in correcting aberration by changing the curvature of the corrector and the secondary independently. Specifically it allows the designer to aspherize the secondary to provide a much wider flat field than traditional spot Maksutovs, with less off-axis coma. Mounting

2925-457: The integral and the supremum can be interchanged with the result being finite if either one is finite. Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational. (A) For a non-decreasing and bounded-above sequence of real numbers

2990-436: The limit lim n → ∞ a n {\displaystyle \lim _{n\to \infty }a_{n}} exists and equals its supremum : (B) For a non-increasing and bounded-below sequence of real numbers the limit lim n → ∞ a n {\displaystyle \lim _{n\to \infty }a_{n}} exists and equals its infimum : Let {

3055-696: The mass and "cool-down time" of a full-aperture corrector. It has the drawbacks of an open, unsealed tube and requires a spider assembly to hold the secondary mirror and corrector, which inevitably affects image quality through diffraction artifacts. Also since the light passes through the corrector twice, the number of surfaces involved is increased, making it difficult to achieve good aberration correction. Sub-aperture corrector Maksutovs are currently manufactured by Vixen telescopes , their VMC (Vixen Maksutov Cassegrain) models. Maksutovs optics can be used in Newtonian configurations that have minimal aberration over

3120-425: The monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions 0 ≤ f 1 ( x ) ≤ f 2 ( x ) ≤ ⋯ {\displaystyle 0\leq f_{1}(x)\leq f_{2}(x)\leq \cdots } , taking

3185-436: The optical elements can be permanently fixed in alignment and the tube assembly can be environmentally sealed, the design is extremely rugged. That makes it ideal for tracking, remote viewing , and radar calibration / boresighting , where instruments are subjected to severe environments and high g-forces . The Rutten Maksutov–Cassegrain (also called a Rumak or Sigler Maksutov ) has a separate secondary mirror mounted on

3250-518: The outcome of the theorem, note that since μ ( N ) = 0 , {\displaystyle {\mu (N)=0},} we have, for every k , {\displaystyle k,} provided that f {\displaystyle f} is ( Σ , B R ≥ 0 ) {\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})} -measurable. (These equalities follow directly from

3315-437: The pointwise supremum is a ( Σ , B R ¯ ≥ 0 ) {\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})} -measurable function and Remark 1. The integrals and the suprema may be finite or infinite, but the left-hand side is finite if and only if the right-hand side is. Remark 2. Under

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3380-527: The possibility of a 'folded' Cassegrain -type construction with a secondary silvered "spot" on the convex side of the meniscus facing the primary mirror . He thought this would create a sealed and rugged optical system suitable for use in schools. This design appeared commercially in Lawrence Braymer's 1954 Questar telescope and in PerkinElmer designer John Gregory 's competing patent for

3445-598: The prices down even further. Many manufacturers currently produce Maksutov–Cassegrains, such as Explore Scientific , Intes, Intes-Micro, LOMO , Orion Optics , Telescope Engineering Company (TEC), Vixen , the Meade Instruments 's ETX line, and the Synta Taiwan produced Celestron , Sky-Watcher and Orion Telescopes lines. The spot Maksutov–Cassegrain design has been used extensively in military , industrial , and aerospace applications. Since all of

3510-733: The proof may skip the intermediate results below. We need three basic lemmas. In the proof below, we apply the monotonic property of the Lebesgue integral to non-negative functions only. Specifically (see Remark 4), lemma 1 . let the functions f , g : X → [ 0 , + ∞ ] {\displaystyle f,g:X\to [0,+\infty ]} be ( Σ , B R ¯ ≥ 0 ) {\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})} -measurable. Proof. Denote by SF ⁡ ( h ) {\displaystyle \operatorname {SF} (h)}

3575-410: The rows are weakly increasing and each is bounded a i , k ≤ K i {\displaystyle a_{i,k}\leq K_{i}} where the bounds are summable ∑ i K i < ∞ {\displaystyle \sum _{i}K_{i}<\infty } then, for each column, the non decreasing column sums ∑ i

3640-403: The same term [REDACTED] This disambiguation page lists articles associated with the title MCT . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=MCT&oldid=1173624668 " Category : Disambiguation pages Hidden categories: Short description

3705-487: The secondary on the corrector also limits diffraction spikes. This version is named after the work of Dutch optical designer Harrie Rutten . Maksutov noted in his designs that instead of using a full-aperture corrector, a small sub-aperture corrector could be placed in the converging light cone of the primary mirror and achieve the same effect. In the 1980s Dave Shafer and Ralph W. Field came out with sub-aperture Cassegrain designs based on this idea. The design reduces

3770-479: The sequence { f n } {\displaystyle \{f_{n}\}} to pointwise non-decrease almost everywhere causes its pointwise limit f {\displaystyle f} to be undefined on some null set N {\displaystyle N} . On that null set, f {\displaystyle f} may then be defined arbitrarily, e.g. as zero, or in any other way that preserves measurability. To see why this will not affect

3835-442: The set { + ∞ } {\displaystyle \{+\infty \}} and all Borel subsets of R ≥ 0 . {\displaystyle \mathbb {R} _{\geq 0}.} Let ( Ω , Σ , μ ) {\displaystyle (\Omega ,\Sigma ,\mu )} be a measure space , and X ∈ Σ {\displaystyle X\in \Sigma }

3900-492: The set function defined by ν A ( S ) = μ ( A ∩ S ) {\displaystyle \nu _{A}(S)=\mu (A\cap S)} is countably additive for all A ∈ Σ {\displaystyle A\in \Sigma } . But this follows directly from the countable additivity of μ {\displaystyle \mu } . Lemma 3. Let μ {\displaystyle \mu } be

3965-589: The set is unbounded. An important use of the extended reals is that any set of non negative numbers a i ≥ 0 , i ∈ I {\displaystyle a_{i}\geq 0,i\in I} has a well defined summation order independent sum where R ¯ ≥ 0 = [ 0 , ∞ ] ⊂ R ¯ {\displaystyle {\bar {\mathbb {R} }}_{\geq 0}=[0,\infty ]\subset {\bar {\mathbb {R} }}} are

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4030-1142: The set of simple ( Σ , B R ≥ 0 ) {\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})} -measurable functions s : X → [ 0 , ∞ ) {\displaystyle s:X\to [0,\infty )} such that 0 ≤ s ≤ h {\displaystyle 0\leq s\leq h} everywhere on X . {\displaystyle X.} 1. Since f ≤ g , {\displaystyle f\leq g,} we have SF ⁡ ( f ) ⊆ SF ⁡ ( g ) , {\displaystyle \operatorname {SF} (f)\subseteq \operatorname {SF} (g),} hence 2. The functions f ⋅ 1 X 1 , f ⋅ 1 X 2 , {\displaystyle f\cdot {\mathbf {1} }_{X_{1}},f\cdot {\mathbf {1} }_{X_{2}},} where 1 X i {\displaystyle {\mathbf {1} }_{X_{i}}}

4095-413: The supremum of the partial sums if and only if the partial sums are bounded. For sums of non-negative increasing sequences 0 ≤ a i , 1 ≤ a i , 2 ≤ ⋯ {\displaystyle 0\leq a_{i,1}\leq a_{i,2}\leq \cdots } , it says that taking the sum and the supremum can be interchanged. In more advanced mathematics

4160-539: The upper extended non negative real numbers. For a series of non negative numbers so this sum coincides with the sum of a series if both are defined. In particular the sum of a series of non negative numbers does not depend on the order of summation. Let a i , k ≥ 0 {\displaystyle a_{i,k}\geq 0} be a sequence of non-negative real numbers indexed by natural numbers i {\displaystyle i} and k {\displaystyle k} . Suppose that

4225-482: Was the first-published meniscus telescope design, and was published in the widely-read Journal of the Optical Society of America . This led to professional and amateur designers almost immediately experimenting with variations, including Newtonian , Cassegrain , and wide-field camera designs. There are many Maksutov designs that use a Cassegrain configuration, mounting a convex secondary mirror near

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