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Completion of measures

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#real-analysis #measure-theory #lebesgue-measure
482
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Approximation property of integrable function

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#measure-theory #lebesgue-measure
256
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Proof of factorization lemma

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#real-analysis #measure-theory #borel-measures
92
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Is there a notion of an almost measurable function?

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#measure-theory #lebesgue-integral #lebesgue-measure
3.6k
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If a sequence of independent random variables converges almost surely to a random variable, then that limit is almost surely a constant

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#probability-theory #measure-theory #random-variables #independence #borel-cantelli-lemmas
708
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Counting Measure not $\sum$-finite

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#measure-theory
80
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$X_n$ be a sequence of +ve $L^1$ r.v. converging a.s. to $X\in L^1$ and $E(X_n) \to E(X)$ . Then $E(X_nY)\to E(XY)$ for any bounded r.v. $Y$

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#probability-theory #measure-theory #random-variables #measurable-functions
2.4k
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The inverse image of a measurable set under a measurable function is measurable?

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#measure-theory #cantor-set #measurable-functions
112
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$X_n$ be a sequence of $L^1$ random variables converging a.s. to $X$ with $\inf_{n\ge 1} E(X_n) >-\infty$. To show $E(|X|)< \infty$

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#probability-theory #measure-theory #random-variables #lp-spaces #measurable-functions
115
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Strong convergence in $L^1$, $\mathbb{R}^d$ with $d>1$

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#real-analysis #measure-theory #lebesgue-measure #product-measure
165
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Outer measure must be defined with countably infinite cubes.

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#real-analysis #measure-theory #proof-verification
31
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How do I prove the measurability of the $j$-th smallement element mapping function?

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#measure-theory #order-statistics
190
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Can this vague feeling about entropy be made precise?

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#measure-theory #ergodic-theory #entropy
202
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Intuition non-measurability of Vitali-Set

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#measure-theory #lebesgue-measure #intuition
134
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How much can the density of a continuous measure be zero?

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#real-analysis #probability-theory #measure-theory #lebesgue-measure

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