# 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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### Holder's inequality in the case of $L_1$ and $L_{\infty}$ norm

I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \begin{equation} \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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### clear example of sigma algebra generated by r.v [duplicate]

I can't understand how a sigma algebra generated by random variable. can you more explain and give me some examples. for example in coin toss we have a sample space and sigma algebra such that it's ...
2answers
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### Relationship between deterministc function of random variables

Given a discrete $P(X,Y,Z)$ let's call $\Omega$ the set of all deterministic functions $f: XYZ \rightarrow W$ and $\Omega'$ the set of all deterministic functions $f': XY \rightarrow V$. Is it correct ...
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### Show $X_n \to 0$ in probability

I am asked to show : Let $X$ be a real-valued random variable on $(\Omega, F , P)$ and define $X_n(\omega) = nX(\omega)$ if $n<X(\omega)\le n+1$ and $0$ if else. Prove that $X_n \to 0$ in ...
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### Product of two probability density function

Suppose $f$ and $g$ are two probability density functions. I have seen economists use $\int f(x)g(x) dx$ as some kind of similarity measure. For example, Jaffe (1986) uses sum of product of two ...
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### Independence of a statistic from size of sample

If you have some (non-constant) statistic $T(X_1, X_2,...,X_n) = f(X_1,X_2,...,X_n)$, is it independent of the number of elements in the sample $(X_1, X_2, ..., X_n)$? That is, is it independent of $n$...
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### Difference between measure and metric in context to information and mutual information

In measure theory, a measure is intrinsic property of the set which informs about the size of the set. On the otherhand an information say a mutual information is a diminishing return property, that ...
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### General Questions on Measure Theory [duplicate]

The notion I have received while trying to study Measure theory that it is helpful when we move from discrete domain to continuous domain because in discrete domain we can count individual entities ...
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### A local base for space of probability measures with Prohorov metric [closed]

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
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### This mixed-type family of random variables has no dominating measure, so a likelihood function can't be defined?

Let $$X = \begin{cases}\theta & \text{with probability 1/2}\\ Z\sim N(0,1) & \text{with probability 1/2.} \end{cases}$$ Here, $\theta\in\mathbb{R}$ is the parameter to be estimated. It ...
1answer
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### Why is measure theory needed to understand continuous random variables and probability density functions in particular?

Prefacing the question with the fact that I have no knowledge of measure theory. I would prefer a conceptual answer, as there already many mathematical ones. Also, why don't we need measure theory to ...
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### How to intuitively visualize measure theory concepts via diagrams? [closed]

Visualizing Measure Theory via Diagrams / Drawings / Geometry Hi, I'm starting to learn about Measure Theory in order to study more advanced topics in Data Science. However, the explanations are ...
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### What's the relationship between these two statements?

Suppose I have a set $A\subset R^2$ and an estimator for set $A$ denoted as $\widehat{A}$. Let $\widehat{A}\Delta A$ denote the symmetric difference between set $\widehat{A}$ and set $A$. For some ...
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### Explain the descriptive statistics notion of population (distribution) to a measure theorist

My understanding of probability theory is a rigorous measure-theoretic one and I know almost nothing about descriptive statistics. However, I need to understand some of the commonly used notions in ...
1answer
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### Integral of distribution followed by Bernoulli

I am new to learning probability theory. Here I got confused but I had a sense of feeling it's correct. This is what I saw from the lecture notes. Let $P\sim Bern(p), Q\sim Bern(q)$. Then is the ...
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### When does it matter if random variables are defined on a common probability space?

If I've got two random variables $X$ and $Y$, when does it matter if I think of them both having the same sample space $\Omega$ or not? I sometimes hear phrases like "assuming a common ...
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### De Finetti's Coherence Principle and Frequentist interpretation

So, without proof or citation, I often see that the Coherence Principle by de Finetti does not hold with Frequentist statistics. It is pretty easy to create examples of this fact. The exception ...
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### Which definition of Kullback-Leibler-Divergence for discrete & continuos variables makes sense?

Motivation In variational inference, one tries to approximate a posterior density $p(\theta|y)$ by another,easier, density $q(\theta)$ in terms of the Kullback-Leibler-Divergence D_{KL}(q(\theta)||p(...
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### Conditional probability density from probabilities

I am trying to understand conditional probablility densities in relation to the conditional probablilities. From the Measure-theoretic definition on Wikipedia, if $X$ and $Y$ are non-degenerate and ...
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### Probability measure condition

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $\{A_i\}_{i\in\mathbb{N}}\subseteq\mathcal{F}$ be a countable partition of $\Omega$, where $\mathbb{P}(A_i)=ab^i,\forall i\geq1$. ...