# Questions tagged [measurable-functions]

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### Multiplication with dilations of nonzero measurable function is injective

Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true: Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
• 583
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### Existence of derivative of distribution of exponential family?

Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is ...
233 views

### Preimage of null sets under a monotone increasing function

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
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1 vote
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### $f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
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### Pointwise limit of a "net" of measurable functions is measurable? [closed]

Let $(X, \mathcal{A},\mu)$ be a finite measure space with the $\sigma$-algebra $\mathcal{A}$ and the measure $\mu$. Let $B$ be a separable Banach space. Then, it is well-known from a theorem by Pettis ...
• 2,749
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### A nonlinear mapping on $L^2(S^1)$ that commutes with all translation operators is necessarily measurable?

Let $H:= L^2(S^1)$, where $S^1$ is the circle, and $\tau_a : H \to H$ be the translation operator for each $a \in S^1$: $$(\tau_a f)(x):= f(x+a)$$ Then, it is clear that ...
• 2,749
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### Characterizing functions that are limits of integrable lower-bounded functions

Let $X$ be a separable Hausdorff topological space, endowed with a positive finite Borel regular measure. Consider those (trivially measurable) functions $f : X \to \mathbb R$ such that their ...
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• 851
1 vote
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### The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple ...
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### Characterization of a Bochner/strongly measurable function solely as a random element

Be $(\Omega, \mathcal{A}, P)$ and $E$ a probability space and a Banach space respectively. This paper of G.A. Edgar contains a proof that, for a function $X: \Omega \rightarrow E$, being weakly ...
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### Measurable selection involving measure valued random variable

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{M}(\mathbb{R}^d)$ be the space of finite signed measures on $\mathbb{R}^d$ endowed with the narrow topology (i.e. the ...
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