Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
40,085
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If $A_n$ are measurable, and $\chi_{A_n} \rightarrow f$ in measure, then $f = \chi_A$ for some measurable $A$.
Question: If $A_n$ are measurable, $\mu(A_n) < \infty$, and $\chi_{A_n} \rightarrow f$ in measure, then $f = \chi_A$ almost everywhere for some measurable $A$.
Approach: If $\chi_{A_n} \rightarrow ...
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When is convergence in measure useful?
[Note: By convergence in measure, I mean that $f_n \rightarrow f$ in measure if $\mu(|f_n - f| \geq \epsilon) \rightarrow 0$ for all $\epsilon$. For probabilitists, this is convergence in probability (...
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Elstodt: Maß- und Integrationstheorie, Chapter 1, Exercise 4.2
I am studying Measure Theory using the book by Jürgen Elstrodt and there is one exercise I am struggling with.
The author mentions the following property of $\sigma$-Algebras: $\sigma(\mathfrak{E}) = \...
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Approximation of a measurable set in product space by rectangles from inside
Suppose $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$ are two finite measure spaces and then $(X\times Y, \mathcal{S}\otimes \mathcal{T},\mu\times \nu)$ is the product space.
I know that for any $C \...
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Can there be a Borel measurable bijection between $\mathbb{R}$ and $\mathbb{R}^2$ [closed]
I know a bijection exists between those sets, that it can be constructed quite explicitly and that it can't be continuous. I'm just not sure about the Borel measurability part.
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Extending the defintion of Jordan inner measure in the 'obvious manner'?
Context:
Tao, in Introduction to Measure Theory, defines the Jordan inner measure of any $E$ bounded subset of $\mathbb R^d$ as,
$$m_{*,J}(E):=\sup\limits_{
\substack{
{A\subset E,\;\;}
{A\...
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Translates of a set of positive Lebesgue measure cover $\mathbb{R}$?
Let $E$ be a set of positive Lebesgue measure in $\mathbb{R}$. Does some countable union of translates of $E$ cover $\mathbb{R}$?
My intuition is that $\mathbb{R}$ can be covered with countable ...
- 570
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Infinite Summation of Almost Sure Convergent RVs
Suppose we have random variables $X_{n,i}$ for $n\geq1,i=1,\dots,b_n$. Here we suppose $b_n$ is non-decreasing. We know that for each $i$, the sequence $X_{n,i}$ converges to $X_i$ almost sure.
Now we ...
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weak convergence and pointwise implies $L_p$ convergence
Suppose $f_i \to f$ weakly in $L^p(X, M, \mu)$, $1 < p < \infty$, and that $f_i \to f$ pointwise $\mu$-a.e. Prove that $f_i^+ \to f^+$ and $f_i^- \to f^-$ weakly in $L^p$.
My proof:
Since $f^\pm ...
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When is $\mathbb E[F(S)\mid S=s]= \mathbb E[F(s)]$ true?
Let $S$ be a discrete random variable on a set $\mathcal S$. Moreover, let $F(s)$ be another random variable (say in $L^1$) for each $s\in\mathcal S$.
Now consider some $s\in\mathcal S$. I am looking ...
- 490
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What is this set? Unknown baire generic set
I'm interested in the set given in this question. For clarity, set
$$S_{\epsilon} = \cup_{j = 1}^{\infty} \left( q_j + \frac{\epsilon}{2^j}, q_j - \frac{\epsilon}{2^j}\right)$$
where $\\{q_j\\}$ is ...
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How to read symbol and understand equivalent measure for a measurable space [closed]
referenceI am reading book on Stochastic Calculus by Gregory F Lawler and struggling with symbols. What does symbol Aµ mean and how to read ...
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Subset of index that minimizes a sum of real values
Given a series of real numbers $c_1, \ldots, c_n$ with $ n \in \mathbb{N} $, is there an algorithm or method to find the subset of indices such that the absolute value of the sum of the values within ...
- 1,070
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Question about the proof that $\hat{G} \cong \widehat{L^1}(G)$
Let $G$ be a locally compact, abelian Hausdorff group. Denote by $\hat{G}$ the abelian group of continuous homomorphisms $\chi: G \rightarrow S^1 \subset \mathbb{C}$ and by $\widehat{L^1(G)}$ the set ...
- 1,356
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How do we know the dual pairing between Lp spaces is well defined? [closed]
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
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