Questions tagged [measurable-functions]

For questions about measurable functions.

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Measurable function definition

In my lecture notes for stochastic processes we define measurable functions as Here $\mathcal{A}$ is the $\sigma-$algebra generated by $\Omega$. However, when I read on other sites (e.g. Wolfram), ...
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22 views

Sheldon Axler Measure Integration Real Analysis Section 2E Exercise 14

As you can tell from the title, this is an exercise from Axler's measure theory book, and I am struggling with the following problem: Suppose $b_1,b_2,\dots$ is a sequence of real numbers. Define $f: \...
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1answer
17 views

Using Egorov's Theorem when the sequence of functions converges pointwise to infinity

I need help with the following question: Suppose $(X, \mathcal{S}, \mu)$ is a measure space with $\mu(X) < \infty$. Suppose $f_1,f_2,\dots$ is sequence of $\mathcal{S}$-measurable functions from $X$...
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Measure, Integration & Real Analysis Sheldon Axler Section 2B Exercise 16

I apologize for the title in advance, but this is a long problem. So, here it is: Suppose $\mathcal{S}$ is a $\sigma$-algebra on a set $X$ and $A \subset X$. Let $$\mathcal{S}_A = \{E \in \mathcal{S} :...
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60 views

$f$ continuous $\Rightarrow$ $f$ $\mu$ measurable?

Let $f: X\to [0,\infty]$ be a continuous function and $\mu$ an outer measure on X. For a continuous function I can split it in a sum of step functions, so $$ f(x) = \sum\limits_{i=1}^\infty s_i \chi_{...
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1answer
30 views

$X=\mathbb{E}[X\mid\mathcal{F}] $ implying $\mathcal{F}$-measurability of $X$

For a random variable $X$ we know, that if $X$ is measurable w.r.t some filtration $\mathcal{F}$ it satisfies $X=\mathbb{E}[X\mid\mathcal{F}]$. I wonder whether one can reverse this property and state,...
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Saks definition of the integral

Let $(X,\mathcal{A},\mu)$ be a measure space and $f:X\to [0,\infty]$ be a function. Define the integral of $f$ by $$L(f)=\sup\bigg\{ \sum_{k=1}^n\inf\{f(x):x\in E_k\}\mu(E_k):\{E_k\} \text{ is a ...
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Sufficient condition for measurability

Let $f$ be a real-valued function defined on a measurable set $E$. Show that the measurability of set {$x$ : $f(x)$ = c} , where c $\in$ $\Bbb R$, is not sufficient for $f$ to be measurable.
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$\mu$-measurable function $f$ and function $\bar{f}$

$(X,A,\mu)$ complete measure space. $f$ $\mu$-measurable on $X$. Then every function $\bar{f} = f$ $\mu$-almost everywhere $\mu$-measurable. This is written in my script. In my understanding: $D, \...
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Are Banach limits of measurable functions measurable?

Let $(X, \mathcal{A}, \mu)$ be a probability space, and let $(f_k)_{k = 1}^\infty$ a sequence in $L^\infty$ such that $\|f_k\|_\infty \leq 1$. Let $L \in \left(\ell^\infty\right)^*$ be a Banach limit, ...
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Borel $\sigma$-algebra $\mathcal{Bor}(\mathbb{R}^d)$ [closed]

Let $\mu$ be a measure on the Borel $\sigma$-algebra $\mathcal{Bor}(\mathbb{R}^d)$ where $d\geq 1$, what is the measure of a single point x of $\mathbb{R^d}$ in this case?
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Explanation of the definition of measurable function

I have just read the definition of measurable function, Which says a function $f$ from $E$ to extended real line is said to be measurable if For all $a$ belongs to $\mathbb{R}$ The set $E(f>a)$ is ...
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1answer
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How to show that $D$ is a Borel measurable set and $D_{f}$ is a Borel function.

How to show that $D_{f}$ is a Borel measurable function. Well I have one Lipschitz function $f:\Bbb{R}^{n}\to \Bbb{R}$ and I want to proof that $D_{f}:D\to L(\Bbb{R}^{n},\Bbb{R})$ is Borel function, ...
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37 views

Example of Non-Measurable Bijection from $\mathbb R \to \mathbb R$

The title is fairly self explanatory. I am looking for an example of a function from $\mathbb R\to\mathbb R$ that is bijective, but not measurable with respect to Lebesgue measure, if such an example ...
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1answer
62 views

Proof that $\int_X K(\cdot,y)f(y) \, d\mu(y)$ is measurable if $K, f\in L^2$?

Let $(X,\mathscr A,\mu)$ be a $\sigma$-finite measure space, $K\in L^2(X^2,\mathscr A \otimes \mathscr A,\mu\otimes\mu),$ and $f\in L^2(X,\mathscr A,\mu).$ I am trying to prove that $F(x)=\int_X K(x,y)...
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Expectation maximizers on a sigma algebra

Let $S$ be a set endowed with a sigma algebra $E$ and $\mu$ an atomless measure on $S$. Let $X$ be a finite set of measurable functions $f: S \rightarrow \mathbb{R}$. I want to prove the following: ...
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32 views

Measurable function such that $f(af(x))$ is not measurable for any $a\ne0$

I know there exist (Lebesgue) measurable functions $f:\Bbb R\to\Bbb R$ such that $f\circ f$ is non-measurable (Here can be found an example). Moreover, on can adapt this construction to find, for any $...
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Lusin’s Theorem for Polish spaces with infinite Radon measure

I’m working on the following exercise in Klenke’s Probability Theory: A Comprehensive Course (Exercise 13.1.3), which asks us to prove the following generalization of Lusin’s Theorem: Let $\Omega$ be ...
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59 views

If $f=g$ almost everywhere, then $\int_E f = \int_E g$.

One task in the book Analysis 2 by Tao is to prove the following Proposition 8.2.6: Let $(E,\mathcal{E},\mu)$ a measure space and $f,g \colon E \to [0,\infty]$ $\mathcal{E}$-measurable functions on $E$...
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$\int_K |u_n-u|^{p-\epsilon} \to 0$ implies that $\int_K|u_n-u|^p \to 0$

Let $K$ be compact set on $\mathbb{R}^n$ with non-empty interior. Assume that $p \geq 2 $ and that it is given a sequence of measurable functions $\{u_n : K \to \mathbb{R}\} \subset L^{p-\epsilon}(K)\...
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How to show that a set is a sigma algebra

I need help with this problom: Let $A={\{E\in \mathcal B(R) | E = -E}\}$ That is the collection of symmetric borel measurable sets with respect to origin I have to show that A is $\sigma-algebra$ ...
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A function that is measurable on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$

This seems to be a result that is used often, which I am having difficulty proving. There may be a more general statement, but I'll write out the statement that I was trying to prove: Let $f:(\mathbb{...
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1answer
38 views

Checking the measurability of a function

I have to check check the measurability of such a function. $f(x):\mathbb{R} \rightarrow \mathbb{R}$ $f(x) = \left\{ \begin{array}{ll} 4x-3 & \textrm{when $x \in V$}\\ 0 & \textrm{when $x \in ...
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1answer
69 views

Show that $f$ is a Borel measurable function

Let $f: \mathbb{R} \to \mathbb{R} $ be a function with $$ x \mapsto \begin{cases} x, \quad &\text{if} \quad x \in \mathbb{Q} \cap [0,1] \\ 1-x, \quad &\text{if} \quad x \in (\mathbb{R} \...
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38 views

Independence of a finite number of random variables.

Let's fix the probability spaces $(\Omega,\mathcal{F},P)$ and $(\Omega',\mathcal{F}',P')$, where $\mathcal{F}$ is a sigma algebra of $\Omega$ and $\mathcal{F}'$ is a sigma algebra of $\Omega'$. ...
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1answer
39 views

Measure preserving transformation exercise

I'm trying to solve this exercise but I need some hints because my teacher didn't give me the theory necessary to solve it. Let $(X, M, \mu)$ be a measurement space such that $\mu (X) = 1$. Suppose ...
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Does progressive measurability of $f(X_t,t)$ imply product measurability of $f$?

Problem. Consider a filtered probability space $(\Omega,\mathcal F,(\mathcal F_t)_{t\in [0,T]},\mathbb P)$ and a function $f:\mathbb R^d\times [0,T]\to \mathbb R^d$. Assume $f(X_t,t)$ is progressive ...
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28 views

Prove $A = \{x ∈ X : f(x) ≥ a\}$ has finite measure [duplicate]

Let $(X, \mathcal{A}, \mu)$ be a measurable space. I want to prove that is $f: X \longrightarrow \mathbb{R}$ is measurable, non-negative and its integral on $X$ is finite, then $\forall a>0$ the ...
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40 views

Complement of a non-measurable set

I want to prove that the complement of a non-measurable set is also not measurable. Let $N \subset \mathbb{R}$ be non-measurable. Then $N \notin \sigma$-Algebra. Sps. $\mathbb{R}$ \ $N$ is measurable, ...
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70 views

Integral of a measurable function over N

I want to calculate the integral of the function $a:\mathbb{N}\to\mathbb{R}$, $a(n)= \frac{(-1)^n}{2^n}$ over the set $\mathbb{N}$, with the cardinal measurement. I've seen how to calculate this type ...
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1answer
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Any measurable function $f:X \to \mathbb R, \Sigma=\{X,\emptyset\}$(where $X$ is a non empty set) must be constant.

Any measurable function $f:X \to \mathbb R, \Sigma=\{X,\emptyset\}$(where $X$ is a non empty set, $\Sigma$ is a sigma algebra on $X$) must be constant. My attempt:- We know that $f:X \to \mathbb R$ is ...
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Composition of generic measurable functions.

Consider $(X,M_X),(Y,M_Y),(Z,M_Z)$ measurable spaces, namely $M_X,M_Y,M_Z$ are $\sigma$-algebras. Consider $f\colon X \to Y$, $g\colon Y \to Z$. Suppose that $f$ is $(M_X,M_Y)$-measurable (namely $\...
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3answers
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Continuous function that is not measurable.

Let $\mathcal{A}$ be a σ-algebra on $\mathbb{R}$ that does not contain all Borel sets (i.e that is, it holds that $\mathcal{B}(\mathbb{R}) \setminus \mathcal{A} \not= ∅)$. Prove that there is a ...
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how to show that if $\int_{*}f=\int^{*}f$ then $f$ is measurable and $\int f=\int_{*}f=\int^{*}f$

how to show that if $\int_{*}f=\int^{*}f$ then $f$ is measurable and $\int f=\int_{*}f=\int^{*}f$. I am working with this: well im try like this: Let $E_{\alpha}=\{x\in X: |f(x)|\leq \alpha\}$ with $\...
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Is the unit circle in $\mathcal{B}(\mathbb{R}^2)$

Is the unit circle in $\mathcal{B}(\mathbb{R}^2)$ ? The definitions I am working with is that $\mathcal{B}(\mathbb{R}^2)$ is the $\sigma$-Algebra generated by open (equivalently closed, or clopen) ...
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1answer
58 views

Induction of topological space by an $L^2$-space

Let $(L^2(\Xi,\boldsymbol{\Xi},\mu;\mathbb{R}),\langle\cdot,\cdot\rangle)$ be a Hilbert space, where $L^2(\Xi,\boldsymbol{\Xi},\mu;\mathbb{R})$ denotes the set of all (equivalence classes of) $\...
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1answer
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Monotone functions are Borel-measurable

I know that there are many similar questions to mine. But this is more a question concerning a certain "logic". I want to show, that every monotone function $f: \mathbb{R} \rightarrow \...
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Induction of measure by a measurable function

Suppose: $(X,\Sigma,\mu)$ is a measure space, $(\mathbb{R},\mathcal{B}_\mathbb{R})$ is the Borel space associated to $\mathbb{R}$ (as topological space), $f:(X,\Sigma)\to(\mathbb{R},\mathcal{B}_\...
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Measurability of |f|^p

In a course of real analysis I am being asked to prove the following: Let $(S,\mathcal{A})$ be a measurable space. Let $f:S \rightarrow \mathbb{R} $ be a measurable function and $p \in (0,\infty)$. ...
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1answer
32 views

Constructing a measure space given the definition of a measurable function

Suppose: $(X,\mathcal{S},\mu)$ is a (complete) measure space, $(\mathbb{R},\mathcal{B}_\mathbb{R})$ is the Borel space associated to $\mathbb{R}$, and $f:(X,\mathcal{S})\to(\mathbb{R},\mathcal{B}_\...
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1answer
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Disproving $ L^2(X, U(\mathbb{C},n)) = U(L^2(X,\mathbb{C}^n))$.

Here $X$ is some measurable space and $U(L^2(X,\mathbb{C}^n))$ denotes the Banach space of unitary automorphisms of the Hilbert space $L^2(X,\mathbb{C}^n)$. Let $U(\mathbb{C},n)$ denote the unitary ...
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1answer
42 views

$f:\mathbb{R} \to \mathbb{R}$ is measurable and $f$ is finite almost everywhere.Then $f=\sum\limits_{k=1}^{\infty}c_k\chi_{E_k}$.

$f:\mathbb{R} \to \mathbb{R}$ is measurable and $f$ is finite almost everywhere.Prove that $f$ can be expressed in the following form $$f=\sum\limits_{k=1}^{\infty}c_k\chi_{E_k}$$ Where $c_k\ \in \...
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Measurability of the density of a transition kernel

Let $P_\theta$ be a family of probability distributions on $(\mathcal{X},\mathcal{F})$ such that $\theta \rightarrow P_\theta(A)$ is measurable $(\Theta,\mathcal{G})$ for all $A \in \mathcal{F}$. Let $...
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1answer
20 views

Pointwise convergence of integrable functions to a real measurable function

I want to prove the following result: If $\left( X, \mathcal{B}, \mu \right)$ is a $\sigma$-finite measure space and $f: X \rightarrow \mathbb{R}$ is a measurable function, then there is a sequence ...
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1answer
32 views

Show $h(\omega_1,\omega_2)=f_1(\omega_1)f_2(\omega_2)$ is $\mathcal{A}_{1} \times \mathcal{A}_{2}$ -measurable.

The question is from a textbook in Chinese that I use for supplementary reading. Let $f_{i}(\omega_{i})$ be measurable on $\sigma$ -finite measure spaces $(\Omega_{i}, \mathcal{A}_{i}, \mu_{i}), i=1,...
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16 views

Preimages of indicator vector

Consider a measurable space $(\Omega,\mathcal{F},P)$, and the random variables $I_A$, $I_B$, such that these are indicator random variables. Now say that $[I_A,I_B]$ is an "indicator vector"....
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1answer
25 views

Show that an integral function is a Borel function

I've been trying to prove that the Hardy-Littlewood Maximal function is a Borel Function for a while now (and was hoping to get some help) and have been stuck on the final step for a while, which is ...
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1answer
39 views

Is measurable function possible in discrete metric space?

The function defined by $f:X{\rightarrow}Y$ is continuous. If $X$ is endowed with discrete metric.  Now,consider characteristic function $\chi_{A}:X{\rightarrow}R$ defined by $\mathcal{X}_E(x) =\begin{...
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24 views

part of the proof of the factorisation lemma

Let $f:(\Omega,\mathcal{A}) \to [0,\infty]$ be measurable (with respect to $\mathcal{A}-\tilde{\mathcal{B}}$) and $f_n \uparrow f$ a sequence of monotonically increasing simple functions that ...
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1answer
27 views

Existence of two measurable functions s.t. $\int_{D}f^3dx = \int_{D}g^4dx = 1$ and $\int_{D}(f+g)^2dx = \pi^3 $

Given $D = \lbrace (x,y) : x^2+y^2 \le 1 \rbrace $, say if we could find two measurable functions $f, g: D \to [0, + \infty]$ s.t. (then give an explicit example) : $$\int_{D}f^3dx = \int_{D}g^4dx = ...

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