# Questions tagged [measurable-functions]

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### Rudin’s PMA, Theorem 11.20

This is the definition which we need for the theorem: (source) 11.19 $\; \;$ Definition $\; \;$Let $s$ be a real-valued function defined on $X$. If the range of $s$ is finite, we say that $s$ is a ...
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### Strong measurability of operator-valued map induced by a kernel

Let $K\in L^2([0,T]^2)$, and for each $t\in [0,T]$, let $\mathcal{T}_t$ be such that for all $f\in L^2([0,T])$, $\mathcal{T}_t f(s)=\int_0^T K(s,t)K(u,t)f(u)\, d u$ for all $s\in [0,T]$. One can show ...
• 12.4k
1 vote
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### open and closed set in $\mathbb{(L_1(\mathbb{R^n})}$

In a test I had was written if $X=(f\in\mathbb{L_1}\mathbb{R^n}|m(f^{-1}((0,\infty))=0)$ is open or closed in $\mathbb{(L_1(\mathbb{R^n})}$. I suspect that this set is none but I am not sure. Is that ...
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### Is the following characterisation of measurable functions true?

I am self studying measure theory.I measure theory it is often important to check if a function is measurable.If the function is continuous then it is measurable of course.But if the function is not ...
• 2,007
1 vote
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### Intuition behind the proof that pointwise limit of measurable function is measurable.

I am self-studying measure theory.There is a very important theorem in measure theory which says that a pointwise limit of measurable functions is measurable.Now,I understand the proof but do not get ...
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### How do I show that a function is measurable with respect to some $\sigma$-algebra?

The definition I have in my notes is as follows: Given a $\sigma$-algebra $\mathcal{G} \subseteq \mathcal{F}$, we say that $X$ is $\mathcal{G}$-measurable or measurable with respect to the $\sigma$-...
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### Characterization of measurable functions from cocountable sigma algebra

Suppose $X$ is a nonempty set and $S$ is the sigma-algebra on $X$ consisting of all subsets of $X$ that are either countable or have a countable complement in $X$. Give a characterization of $S$ ...
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### Composition by exponentiation of measurable functions

A measurable function is defined the following way: $f:X \rightarrow [-\infty, +\infty]$ such that $f^{-1}((-\infty, a))\in \mathcal{M} \ \ \forall a \in \mathbb{R}$ Then, there is this following ...
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### Measurability and equality of functions on product measure space

Let $(\Omega_1,\mathcal{A}_1,\mu_1)$, $(\Omega_2,\mathcal{A}_2,\mu_2)$ be two finite measure spaces and denote by $(\Omega_1 \times \Omega_2,\mathcal{A}_1 \otimes \mathcal{A}_2,\mu_1 \otimes \mu_2)$ ...
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### Approximation by simple functions for $f\in L^1(\mu)$

I'm solving a problem about measure space. Let $(X,\mathcal{M}, \mu)$ be a measure space, and let $f \in L^1(\mu)$. Prove that, for all $\epsilon >0$, there exist a simple measurable function $s$ ...
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### question about limit of integral

Let $(X,\mathcal{M}, \mu)$ be a measure space such that $\mu(X) < \infty$ and let $f,g,h \in L^1(\mu)$. For $n \in \mathbb{N}$, define $$B_n = \{x \in X: |f(x)| +|g(x)| \leq n \} \in \mathcal{M}$$...
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1 vote
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### Criterion for a function to be Lebegsue measurable.

We know that a function from $\mathbb R$ to $\mathbb R$ is continuous iff the graph can be drawn without lifting the pen. I want to know if there is a similar intuitive characterisation for measurable ...
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