# 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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### Sigularity of the Covariance Matrix and Absolute Continuity of the Distribution [duplicate]

Suppose that X is a multivariate normal vector with the covariance matrix $\Omega$. As we know, if $\Omega$ is singular, the density of X is not defined in the usual way (because the denominator is ...
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1 vote
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### Mathematical knowledge needed for learning upper level statistics

im a math student who wants to study statistics in depth for a master degree(maybe doctor degree). And I was wondering what kind of math knowledge might be needed for upper level statistics. I wanna ...
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### If $F_X, F_Y$ agree for all $x \in \mathbb{R}$, Do their distributions $\mu_X, \mu_Y$ agree on $\mathcal{B}$?

Let random variables X and Y be defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume their distribution functions $F_X$ and $F_Y$ agree for all $x \in \mathbb{R}$. How ...
1 vote
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### What is the measure-theoretic definition of the condition probability?

Let $(S, \mathcal{S}, \Pr)$ be a probability space. Let $(\mathcal{X}, \mathcal{B})$ be a standard Borel space and a measurable map $X : S \to \mathcal{X}$. Let $(\Omega, \tau)$ be a standard Borel ...
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1 vote
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### Singular distribution via push forward

Suppose $X$ is a multivariate normal distribution in $\mathbb{R}^3$ with a covariance matrix whose rank is 2. Therefore, it is a singular distribution. Is it possible to represent it as a push forward ...
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### Probability of getting a rational number for continuum trials

As far as I know, the probability of getting a rational number from the interval $(0, 1)$ is zero if we follow the Lebesgue measure. Still, it doesn't mean it's impossible. Now, suppose we have a ...
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### Regular conditional distribution vs conditional distribution

What is the difference between the concept of the regular conditional distribution and the concept of the conditional distribution? Why do we need these two different concepts? Under which ...
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1 vote
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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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