# Questions tagged [measurable-functions]

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### Prove that a continuous and $\mathcal{F}_t$ adapted process is progressively measurable

I would like to show that if $X_t$ is a continuous and adapted stochastic process (real valued) then it is progressively measurable. Here is my attempt : consider $B\in\mathcal{B}(\mathbb{R})$. Then ...
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### Bijection that takes continuum cardinal and null set to positive measure set

I've been dealing with the following problem: Suppose $f:\mathbb{R} \to \mathbb{R}$ is a bijection that verifies $m(f^{-1}(C)) > 0$ for $C$, a set with the continuum cardinal and $m(C)=0$, where $m$...
1 vote
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### Prove that $\{X_n\to X\}=\Omega\setminus \big\{\limsup_{n\to\infty}|X_n-X|>0\big\}$

Let $\Omega$ be measurable space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of real random variables. Denote by $\color{red}{\{X_n\to X\}}$ the set that contains exactly all elements $\omega\in \Omega$...
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### $f_n \to f$ almost uniform if and only if $\lim_{n \to \infty} \mu ( \cup_{m \geq n} |f_m(x) - f_n(x)| \geq \epsilon ) = 0$

Show that $f_n \to f$ almost uniform if and only if for all $\epsilon >0$ $\lim_{n \to \infty} \mu ( \cup_{m \geq n} |f_m(x) - f_n(x)| \geq \epsilon ) = 0$ I have showed the first direction, but ...
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### A Criterion that helps verify if a function $f:\Pi _{i=1}^nX_i\to Y$ is measurable

Let $\{X_i\}_{i=1}^n$ be a collection of measurable spaces and $Y$ a measurable space. Is there a criterion that helps verify if a function $f:\Pi _{i=1}^nX_i\to Y$ is measurable? I know that a ...
1 vote
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### On the definition of locally integrable functions on an abstract measure space.

Recently I've been able to find a very cheap copy of the nice monograph , where I can find (chapter 10, §1, p. 163) the following general definition of Locally integrable function (respect to a ...
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### Counterexample to Egorov for functions valued in non-separable metric space

A general form of Egorov (e.g. https://www.ime.usp.br/~glaucio/mat6704/textos/GMTLecureNotes.pdf) states that: Egorov Theorem : Let $\mu$ be an outer measure on the set $X$ and $(Y,d)$ a separable ...
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### Topology of convergence in measure is not compatible with the vector space structure of measurable functions

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the ...
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### Does Amann's Theorem 1.4 about $\mu$-measurability extend to metrizable topological groups?

Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space and $(E, | \cdot |)$ a Banach space. Let $f:X \to E$. We recall some definitions at page 62 of Amann's Analysis III. $f$ is ...
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### If $f \in L^0 (X, L^0 (Y))$, then $f \in L^0 (Z)$

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $\mu(X) < \infty$, $(E, | \cdot |)$ a ...
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### If for a.e. $x \in X$ the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^0 (Y)$, then $(f_n)$ is Cauchy in $L^0 (Z)$

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of ...
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### If $\| g_n \|_{L^0(X)} \to 0$ then $\| f_n \|_{L^0 (Z)} \to 0$

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of ...
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### If $\nu(Y) < \infty$ then $F: X \to L^0(Y), x \mapsto f(x, \cdot)$ is $\mu$-measurable

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the ...
20 views

### Convergence in a complete measure implies a.e. convergence for a subsequence

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple functions from $X$ ...
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1 vote
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### Whether the function $(x,x')\mapsto\rho\big(f(x),f(x')\big)$ is Borel for a Borel map $f\colon (X,d)\to (Y,\rho)$ with $(Y,\rho)$ not being separable?

Let $(X,d)$ be a compact metric space, let $(Y,\rho)$ be an arbitrary metric space. Let $\mathcal{B}(\mathbb{R}),\mathcal{B}(X),\mathcal{B}(Y),\mathcal{B}(X\times X), \mathcal{B}(Y\times Y)$ denote ...
1 vote
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### Continuous extension of compactly supported continuous functions

Suppose $D\subset \mathbb{R}^d$ is a path-connected compact (closed and bounded) domain. Let $C\subset D$ be a compact subset. If the restriction $f|_C$ of a funciton $f: D\to \mathbb{R}$ is ...
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### $f: E \rightarrow \mathbb{R}$ is $\mathcal{E}$-measurable if and only if $f^{+}$ and $f^{-}$ are $\mathcal{E}$-measurable

Consider the measurable spaces $\left( E,\mathcal{E} \right)$ and $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$. Prove that $f:E\rightarrow \mathbb{R}$ is $\mathcal{E}$-measurable $\iff$...
1 vote
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### Measurability with respect to complete filtration

I don't know if this is a difficult question, but I have absolutely no intuition about it. Let us consider a probability space $(\Omega, \mathcal F, \mathbb P)$ supporting a $\mathbb R-$valued ...
1 vote
Consider an integer $k\in\mathbb{N}$. Let $V_{k}, V, H$ be distinct Hilbert spaces with identical inner products and, therefore, the same norms. We have: $$V_{k},V\subset H$$ In this context, I ...
The setting for DCT is some fixed measure space $(X,M,\mu)$ Proof. $f$ is measurable (perhaps after redefinition on a null set) by Propositions 2.11 and 2.12, and since $\left |f \right | \leq g$ a....