# 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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### Confirmation of MGF of Shifted Exponential Distribution

Consider the probability distribution $D \sim X - a$ where $X \sim \text{Exp}(\lambda)$. My task is to find the MGF of probability distribution $D$. I think I have a solution but it contradicts what I ...
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### Two Different Proofs of Continuous Mapping Theorem

The theorem to prove is that if $X_n$ converges weakly to $X$, and $P(X \in D_g) = 0$ where $D_g$ is the set of discontinuity of $g$, then $g(X_n)$ converges weakly to $g(X)$. In Durrett, this is ...
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### Inverting a characteristic function if the integral of the modulus of the cf is infinity

I'm reading a lecture slide that starts by asking if there's a way to invert a characteristic function $\psi_X$ if $\int|\psi_X(t)|~\mathrm{d}t = \infty$. From my reading, the slide then provides a ...
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### How do we define the pdf in the multi-variate case and compute expectations?

Apologies if this is a very simple question but trying to work through a result in a paper made me realize I missed something a bit fundamental in my undergrad probability and analysis courses. Lets ...
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### Interchange of integral and differentiation [migrated]

Say I have 2 continuous Bivariate distributions( for example bivariate log normal and bivariate weibull) which have joint density functions as: $f_1(x,y,\theta)$ and $f_2(x,y,\alpha).$ Now I want to ...
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### 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 ...
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### What does it mean to condition on a sub-sigma-field? [duplicate]

I am studying measure-theoretic probability independently. An earlier course on probability dealt with conditioning on an event and on a random variable. These lecture notes discuss $P(A|C)$ where $C$ ...
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### 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 ...
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### 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 ...
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### How do we choose the sigma algebra for a specific experiment?

I am trying to understand the connect between probability and measure theory. I get some basic things about measure theory. But what I still don't get is how do we choose the sigma algebra for our ...
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### Quick question about density with respect to product measure

Suppose we got 2 non indepenendent random variables $X_1:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_1,\mathcal{B}_1)$ and $X_2:(\Omega,\mathcal{A})\rightarrow (\mathcal{X}_2,\mathcal{B}_2)$ with ...
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### Correct conditional expectation with respect to a different measurable space

Suppose we've got random variables $X_1:(\Omega_1,\mathcal{A}_1)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R})),X_2:(\Omega_2,\mathcal{A}_2)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$ on a ...
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### An extremely basic question about statistics

We do statistics on data which we assume to be outcome of random trials. The question however, is -- are we assuming that the outcome can be represented as a random variable? The outcome of an ...
1 vote
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### Precise Definition of $\mathbb{E}[X\mid \sigma(A)]$ Conditional Expectation of Random Variable given Sigma Algebra generated by a set

I want to define precisely, exhaustively and constructively the conditional expectation of a random variable given the sigma algebra generated by a set. This question has some discussion on it but the ...
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### Computing certain integrals with a particular distribution function

Let $(\xi_n, n \in \mathbb{N})$ be a sequence of random vector such that $G_n$ is the distribution function of $\xi_n$. Let $\big(\psi_{j}^n \big)_{(j,n) \in \mathbb{N}^2}$ be a double sequence such ...
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### How to formally define a conditional distribution conditioning on an event of probability zero?

Given $[X,Y]\sim N(0,I_2)$, a intuitive guess of the value of $P(X=x|\{X,Y\}=\{x,y\})$, where $\{\}$ means unordered set, is $1/2$ by symmetry. This type of notations is typically applied in ...
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### What is the difference between a probability measure and a probability density function? [duplicate]

During my research, I have repeatedly come across the terms probability measure and probability density function (pdf). I am familiar with the concept of a pdf, but I am not entirely sure how ...
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### Difference in Probability Measure vs. Probability Distribution

I am trying to better understand the Difference in "Probability Measure" and "Probability Distribution" I came across the following link : https://math.stackexchange.com/questions/...
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