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| 5번째 줄: | 5번째 줄: | ||
===말뭉치=== | ===말뭉치=== | ||
# Expected value, which we consider in the next chapter, can be interpreted as an integral with respect to a probability measure.<ref name="ref_93684c84">[https://stats.libretexts.org/Bookshelves/Probability_Theory/Book%3A_Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/03%3A_Distributions/3.10%3A_The_Integral_With_Respect_to_a_Measure 3.10: The Integral With Respect to a Measure]</ref> | # Expected value, which we consider in the next chapter, can be interpreted as an integral with respect to a probability measure.<ref name="ref_93684c84">[https://stats.libretexts.org/Bookshelves/Probability_Theory/Book%3A_Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/03%3A_Distributions/3.10%3A_The_Integral_With_Respect_to_a_Measure 3.10: The Integral With Respect to a Measure]</ref> | ||
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# The following definition reflects the fact that in measure theory, sets of measure 0 are often considered unimportant.<ref name="ref_93684c84" /> | # The following definition reflects the fact that in measure theory, sets of measure 0 are often considered unimportant.<ref name="ref_93684c84" /> | ||
# We also need to extend topology and measure to \( \R^* \).<ref name="ref_93684c84" /> | # We also need to extend topology and measure to \( \R^* \).<ref name="ref_93684c84" /> | ||
| 27번째 줄: | 26번째 줄: | ||
# that satisfies must be a probability measure as defined earlier; that is, it satisfies for all .<ref name="ref_2ab61db3" /> | # that satisfies must be a probability measure as defined earlier; that is, it satisfies for all .<ref name="ref_2ab61db3" /> | ||
# ∞ -\infty is not allowed as a value for a signed measure.<ref name="ref_2ab61db3" /> | # ∞ -\infty is not allowed as a value for a signed measure.<ref name="ref_2ab61db3" /> | ||
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===소스=== | ===소스=== | ||
<references /> | <references /> | ||
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| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q192276 Q192276] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LEMMA': 'measure'}] | ||
2021년 2월 17일 (수) 00:56 기준 최신판
노트
위키데이터
- ID : Q192276
말뭉치
- Expected value, which we consider in the next chapter, can be interpreted as an integral with respect to a probability measure.[1]
- The following definition reflects the fact that in measure theory, sets of measure 0 are often considered unimportant.[1]
- We also need to extend topology and measure to \( \R^* \).[1]
- \(\mathscr S\) is the collection of all subsets of \(S\), and \( \# \) is counting measure on \( \mathscr{S} \).[2]
- \lambda(A_i) \) is simply the length of the subinterval \( A_i \), so of course measure theory per se is not needed for Riemann integration.[2]
- \mu \) is, appropriately enough, referred to as the Lebesgue-Stieltjes integral with respect to \( F \), and like the measure, is named for the ubiquitous Henri Lebesgue and for Thomas Stieltjes.[2]
- A \) is computed by integrating the density function, with respect to the appropriate measure, over \( A \).[2]
- Measure the altitude of the mountain at the center of each square.[3]
- The Lebesgue integral is obtained by slicing along the y-axis, using the 1-dimensional Lebesgue measure to measure the "width" of the slices.[3]
- Measure theory was initially created to provide a useful abstraction of the notion of length of subsets of the real line—and, more generally, area and volume of subsets of Euclidean spaces.[3]
- This means that a measure is any function μ defined on a certain class X of subsets of a set E, which satisfies a certain list of properties.[3]
- "This text succeeds in its aim of providing an introduction to measure and integration that is … accessible to undergraduates.[4]
- The modern treatment of probability is to view it as part of measure theory.[5]
- The axioms of additivity and complementation, which are basic to probability, coincide with the axioms of the Borel-Lebesque measure.[5]
- The aim of the present chapter is to provide the basic tools for acquiring working knowledge of the Lebesgue integral and its generalisations to abstract measure and integration theory.[6]
- A full appreciation of measure theory requires, we believe, some insight into the genesis of the subject.[6]
- The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set.[7]
- It uses a Lebesgue sum where is the value of the function in subinterval , and is the Lebesgue measure of the set of points for which values are approximately .[7]
- It is also possible to take the entire expression ‘ d μ \mathrm{d}\mu ’ as the name of the measure, writing d μ ( A ) \mathrm{d}\mu(A) even where the common notation is μ ( A ) \mu(A) .[8]
- A measure space is a measurable space equipped with a measure.[8]
- that satisfies must be a probability measure as defined earlier; that is, it satisfies for all .[8]
- ∞ -\infty is not allowed as a value for a signed measure.[8]
소스
- ↑ 1.0 1.1 1.2 3.10: The Integral With Respect to a Measure
- ↑ 2.0 2.1 2.2 2.3 The Integral With Respect to a Measure
- ↑ 3.0 3.1 3.2 3.3 Lebesgue integration
- ↑ Measure, Integral and Probability
- ↑ 5.0 5.1 Measure and Integration - an overview
- ↑ 6.0 6.1 The Lebesgue measure and integral (Chapter 2)
- ↑ 7.0 7.1 Lebesgue Integral -- from Wolfram MathWorld
- ↑ 8.0 8.1 8.2 8.3 measure space in nLab
메타데이터
위키데이터
- ID : Q192276
Spacy 패턴 목록
- [{'LEMMA': 'measure'}]