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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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Multidimensional Stieltjes measure functions

In the first chapter of the book "Probability: Theory and Examples" by Rick Durrett, a function $F:\mathbb{R}^n\to\mathbb{R}$ is said to be a Stieltjes measure function if $F$ is non-decreasing, that ...
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Example of a non-measurable maximum likelihood estimator

If you have a measurable parameter space $(\Theta, \mathcal{F})$ and a parametric family of probability measures $(P_\theta)_{\theta \in \Theta}$ on a measurable space $(\mathcal{X}, \mathcal{B})$ ...
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The meaning of distance measure in Additive attention and Dot Product Attention

Currently, I am studying about Transformer (reading this paper; attention is all you need). In the paper, the author said, (3.2.1) Additive attention computes the compatibility function using a ...
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What is the complement of an event in a sigma algebra?

Consider $\Omega=\{1,2,...,6\}$ and $F=\left\{\{1\},\{3\},\{5\}\right\}$. Let $A={1}$. If $F$ is to be a sigma algebra, then necessarily $A^c \in F$. What is $A^c$?
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“Natural” finite measure over continuous probability densities over the interval $[0,a]$ [closed]

I wonder whether there is a "natural" finite measure $\mu$ (such as the Lebesgue-Measure on $\mathbb{R}\cap[0,a]$) over the space of all continous probability density functions on $[0,a]$. EDIT: As ...
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Independence of multivariable function of random variables on the information

It might be this question was asked before, but I still want to request an answer in measure-theoretic framework. Let's define a probability space $\{\Omega, \mathcal{F}, \mathbb{P}\}$ and $\sigma$-...
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Conditional expectation given event and random variable

Let $W$ be a random variable that only takes on the values $1$ or $0$. Let $X$ and $Y$ be two other random variables. I came across the following: $$\mathbb{E}(Y|W=1, X)$$ How is this 'conditional ...
I frequently work with population studies, where, let's say, age and sex are collected for $N=1,000,000$ individuals in California. I might ask a simple question: what is the average age in this ...
The Halmos-Savage theorem says that for a dominated statistical model $(\Omega, \mathscr A, \mathscr P)$ a statistic $T: (\Omega, \mathscr A, \mathscr P)\to(\Omega', \mathscr A')$ is sufficient if (...