Questions tagged [measurable-functions]

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Is a $\mathcal B(R_+) \otimes \mathcal F$-jointly measurable process $X(t,\omega)$ measurable when restricted to $\mathcal B[0,T] \otimes F$

Let $X: (\mathbb R_+ \times \Omega, \mathcal B(\mathbb R_+) \otimes \mathcal F ) \rightarrow (\mathbb R_+, \mathcal B(\mathbb R_+))$ be a jointly measurable map. I would like to prove that the ...
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Matrix function of a measurable map $f$ is again measurable?

Consider a measurable map $f: \mathbb{R} \to \mathbb{R}$. Define the matrix function as the induced map f: H^n \to H^n \\ A = \sum_i \lambda_i e_i e_i^T \mapsto \sum_i f(\lambda_i) ...
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Measurability of $\int_\Omega \varphi(x)u(t,x) \mathrm{d}x$ for $\varphi \in L^1(\Omega)$ and $u$ in a Bochner space

I have a function $u \in L^\infty((0,\infty), L^\infty(\Omega))$ where $\Omega$ is a bounded domain. Take $\varphi \in L^1(\Omega)$ and consider $$f(t) := \int_\Omega \varphi(x)u(t,x) \mathrm{d}x.$$ ...
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Is the given set Borel measurable?

Let $$S=\left \{\sum _{i \geq 1}\frac{x_i}{10^i} \in [0,1]: x_i\in\{0,1,...,9\}\text{ and} \sum _{i \geq 1}\frac{(-1)^{x_i}}{i}\text{ converges}\right \}$$, is the set Borel measurable? Here's my ...
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How to prove $f$ is measurable in $E\subset\mathbb{R}^n$ with following conditions?

I'm stuck in this problem: $f$ is the extended real-valued function in $E\subset\mathbb{R}^n$ and $E$ is measurable. Prove $f$ is measurable if $\forall\varepsilon>0$ , it exists a measurable ...
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On stopping time and filtration containments

This is an attempt to clarify an issue from Section 6 in the proof of Theorem 2.2. Bass, R. F., Uniqueness in law for pure jump Markov processes, Probab. Theory Relat. Fields 79, No. 2, 271-287 (1988)....
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On the measurability of a certain function

Some techniqal question regarding the measurability of a function: Assume $f :\mathbb{R}^n \rightarrow \mathbb{R}$ is measurable. Clearly $(x,y)\mapsto x-y$ is measurable. My question is, is it ...
• 269
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Composition of Lebesgue measurable function and Invertible linear transformation is Lebesgue measurable

Let $f:\mathbb{R}^n\to \mathbb{R}$ is a Lebesgue measurable function and $T\in Gl(n,\mathbb{R})$ then show that $f\circ T$ is also Lebesgue measurable . For Borel measurable functions it's easy to ...
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