# Questions tagged [measure-theory]

A field of mathematics concerning measures, which are functions mapping sets to real numbers that generalize the idea of area and volume. Measure theory underlies modern treatments of probability. Questions about measure theory with a weak connection to probability or statistics may be more suitable for math.stackexchange.com than this site.

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### What is the relationship between convergence in measure and convergence in Kolmogorove distance?

Suppose I have a sequence of distribution $F_1, ..., F_n$ convergence to a limiting distribution $F$ in the sense that \begin{align} \sup ||F_n - F|| \overset{a.s.}{\rightarrow} 0 \end{align} which ...
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### When running an A\B test, do we say we experiment yields two sample spaces, one for each arm, or a single sample space?

Say we have a well setup A\B test that puts half of users in one arm, half in another. Do we describe the sample space as $\Omega_1$ for arm1 and $\Omega_2$ for arm2? Or do we instead just say the ...
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### Calculate the cosine similarity between two probability measures, rather then between two vectors

Introduction. The cosine similarity is often calculated among two vectors, i.e. $C_S(X,Y)$: C_S(X,Y) = \frac{\sum_{i=1}^{n}(X_i,Y_i)}{\sqrt{\sum_{i=1}^{n}X_{i}^{2} \sum_{i=1}^{n}Y_{i}^...
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### Correspondence between the "density function of a probability measure" and the "probability density function" (PDF)

Question. If there is a one to one correspondence between a "borel probability measure" on the line $\mathbb{R}$ and a "cumulative distribution function" (CDF) (please see on page ...
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### Could it make sense to calculate the "distance correlation" among two probability measures (probability distributions)?

Introduction. I post here the same question that I asked on math.stackexchange, since it might be more relevant to this forum: Could it make sense to calculate the "distance correlation" ...
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### Equivalent Formulations of Thompson Sampling

I am studying Chapter 36 Thompson Sampling of the book Bandit Algorithms by Lattimore and Szepesvari. The authors present two equivalent formulations of Thompson Sampling on page 460, and I am having ...
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### Why is this integral not Lebesgue measurable? (and how to rectify)

Suppose I have the Bernoulli random variables $S_t$ for $t=1,\cdots, T$ and for an issue related to my problem, I want to express $P[S_t=s_t\mid S_{t-1} = s_{t-1}]$, by using the law of total ...
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### Definition of exponent measure (extreme value theory)

Let $F$ be a distribution function on $\mathbb{R}^2$, and let $U_i$ be the left continuous inverse of $\frac{1}{1-F_i}$, where $F_i$ is the marginal distribution of $F$. In my textbook, there is the ...
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### How to think of a sub-sigma algebra as a collection of random variables?

Where $\mathcal{H}$ is a $\sigma$-algebra on $\Omega$, section 9.1 here discusses thinking of a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{H}$ "as the collection of all numerical random ...
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Hello I have the assumptions that $E(X) = 0$ and $\operatorname{Var}(X) = {\sigma}^{2}$ . Why does this inequality hold $P(X \geq \lambda) \leq \frac{{\sigma}^{2}}{{\sigma}^{2} + {\lambda}^{2}}$ for $... • 35 1 vote 1 answer 36 views ### Equivalence of Tightness of Seqeuence of CDFs In Durrett, a sequence of cdf$\{F_n\}$is called tight if for all$\epsilon > 0$< there exists$M_\epsilon$such that$\limsup 1-F_n(M_\epsilon)+F_n(-M_\epsilon) \leq \epsilon$. In Rosenthal, a ... • 626 1 vote 0 answers 23 views ### Prove that smallest$\sigma$-field of subsets of$A$containing the open sets in$A$is$\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$[duplicate] Let$A$be a Borel set of$\mathbb{R}$. Then show that the smallest$\sigma$-field of subsets of$A$containing the open sets in$A$is$\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$. I am ... 0 votes 0 answers 88 views ### Prove that smallest$\sigma$-field of subsets of$A$containing the open sets in$A$is$\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$Let$A$be a Borel set of$\mathbb{R}$. Then show that the smallest$\sigma$-field of subsets of$A$containing the open sets in$A$is$\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$. I am ... 1 vote 1 answer 251 views ### Density from characteristic function: Durrett example 3.3.8 and 3.3.9 Letting$\varphi(t)$be the characteristic function for the probability measure$\mu$, we know if$\int \left|\varphi(t)\right|dt < \infty$, then$\mu$has density function$$f(y) = \frac{1}{2\pi} \... • 626 2 votes 1 answer 116 views ### Support of a continuous distribution Suppose I have a continuous random variable$X$on$\mathcal{R}^1$, with CDF$F(\cdot)$and pdf$f(\cdot)$. My understanding is that there are three equivalent definitions of the support of the random ... • 21 0 votes 0 answers 40 views ### From pointwise convergence to uniform: metrics Let$\mu_\theta$be the limit of an empirical measure$\mu_{n, \theta}$.$\theta \in \Theta$and$\Theta$is a compact set. Morever, the maps$\theta \rightarrow \mu_{n, \theta}$and$\theta \...
Let $X$ be a continuous random variable which induces a probability measure on $\mathbb{R}$ denoted by $\mu$. Are there any instances in statistics when we deal with random variables $X$ such that \$\...