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위키데이터

• ID : Q192276

말뭉치

1. Expected value, which we consider in the next chapter, can be interpreted as an integral with respect to a probability measure.
2. # \), counting measure.
3. The following definition reflects the fact that in measure theory, sets of measure 0 are often considered unimportant.
4. We also need to extend topology and measure to \( \R^* \).
5. \(\mathscr S\) is the collection of all subsets of \(S\), and \( \# \) is counting measure on \( \mathscr{S} \).
6. \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.
7. \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.
8. A \) is computed by integrating the density function, with respect to the appropriate measure, over \( A \).
9. Measure the altitude of the mountain at the center of each square.
10. The Lebesgue integral is obtained by slicing along the y-axis, using the 1-dimensional Lebesgue measure to measure the "width" of the slices.
11. 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.
12. 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.
13. "This text succeeds in its aim of providing an introduction to measure and integration that is … accessible to undergraduates.
14. The modern treatment of probability is to view it as part of measure theory.
15. The axioms of additivity and complementation, which are basic to probability, coincide with the axioms of the Borel-Lebesque measure.
16. 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.
17. A full appreciation of measure theory requires, we believe, some insight into the genesis of the subject.
18. The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set.
19. 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 .
20. 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) .
21. A measure space is a measurable space equipped with a measure.
22. that satisfies must be a probability measure as defined earlier; that is, it satisfies for all .
23. ∞ -\infty is not allowed as a value for a signed measure.