Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
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Intuition behind the exponential convergence(e-convergence)
I'm studying a concept called e-convergence for sequences of probability densities. The definition states:
A sequence $(g_n)_{n \in \mathbb{N}}$ in $M_{\mu}$ is e-convergent to $g$ if:
$(g_n)_{n \in \...
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Moment method and central limit theorem
Consider functions $P_n$ depending on a parameter $n$, and a fixed function $\phi$. Consider also a discrete set $D$. Assume we have the convergence
$$
\sum_{d \in D} P_n(d)^k \phi(d) \underset{n \to \...
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Is the $\alpha$-Hausdorff content of a set of diameter $\delta$ equal to $\delta^\alpha$?
Fix $\alpha>0$. The $\alpha$ dimensional Hausdorff content of a bounded set $K$ is given by
$$H^{\alpha}_{\infty}(K)=\inf\left\{\sum_{i=1}^\infty \text{diam}(B_i)^\alpha:K\subset\bigcup_{i=1}^\...
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Weak* topology and probability measures problem
I'm working on a problem related to the weak* topology of probability measures on a metric space and would appreciate your help. Here's the statement of the problem:
Consider $K$ a metric space and $\...
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4
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Confusion on defining uniform distribution on hypersphere and its sampling problem
Fix a dimension $d$. Write $S^{d-1}$ for the surface of a hypersphere in $\mathbb{R}^d$, namely set of all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$ such that $|x|^2 = x_1^2 + \cdots + x_d^2 = 1$. I ...
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theorem radon nikodym
I am reading the proof of the theorem, in Stein's book real analysis (page 290/291) and I cannot understand why when you use the function g (you already know that it exists from a previous theorem) ...
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Find the probability density function of $(B_t^\mu, S_t^\mu)$ [closed]
Let $B=(B_t)_{t\geq 0}$ be a standard Brownian motion and let $B_t^\mu = B_t + \mu t$ be a Brownian motion with drift. Then let $S_t = \sup_{0 \leq s \leq t} B_s$ and $S_t^\mu = \sup_{0 \leq s \leq t} ...
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Deriving an inequality for the integral of maximum indicator functions under measure-preserving transformations
Let's denote the measure space by $(X, \mathcal{B}, \mu)$ and the measure-preserving transformation by $T: X \to X$. Let $A \in \mathcal{B}$ be a measurable set with $0 < \mu(A) < \infty$. Let $...
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If $ x \in \mathbb{R}^n $ fixed, then $ \int_{Q_k}f(x,2^k)dx = f(x,2^k)\mu(Q_k) $. [closed]
I have a question in the following.
$Q_k$ is a open cubes such that $Q_k = \{ x = (x_1,x_2,\cdots, x_n)\in \mathbb{R}^n : a_{k-1} < x_i < a_k\,, i = 1,2, \cdots,n\},$
where $a_0 \in \mathbb{R}$.
...
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There is a unique coupling between a probability distribution $\mu$ and a degenerate distribution $\mu_0$.
By degenerate distribution I intend https://en.wikipedia.org/wiki/Degenerate_distribution.
I cannot see why the set of couplings between some probability measure and a constant would be a singleton. ...
-2
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1
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A measurable bijection between the interval and the square [closed]
Is it possible to find a function $f:[0,1) \rightarrow [0,1)^2$ such that $f$ is bijective and $f$ as well as $f^{-1}$ are measurable with respect to the corresponding Lebesgue measures.
If so how do ...
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Kraus operators
Suppose we have a POVM given by the family of positive, hermitian operators $\{E_i\}_{i\in I} \in \mathcal{H}$.
From the Neimark dilation theorem we know that the given POVM can be obtained from ...
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