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Questions tagged [measurable-functions]

For questions about measurable functions.

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Increasing function is measurable

First of all, I want you to know that I've checked all the questions related to mine and they seem not to be complete, in my opinion. Let me tell you why. Here's the problem: Let $f:\mathbb{R}\to\...
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Pushforward measure

Good evening, Let $\mu$ and $\nu$ two measures on $X$ and $Y$. Do you know when it exists a measurable function $h : X \rightarrow Y$ such as $ \nu = h\text{#}\mu$ with $ h\text{#}\mu(B) = \mu(h^{-1}...
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Proving Borel measurability of a function.

Suppose $f: \mathbb{R}^2 \to \mathbb{R}$ is a function such that $f(x, \cdot)$ is Borel-measurable for each $x$, and $f(\cdot, y)$ is continuous for every $y$. Define $f_n: \mathbb{R}^2 \to \mathbb{...
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Why if $U$ is open in $\overline {\Bbb R}$ then $U\cap \Bbb R$ is open in $\Bbb R$ in this proof?

Consider $f:\mathbb R\to\overline{\mathbb R}$ defined as $f(x)=\begin{cases}\frac{1}{x}, x\neq 0\\ \infty, x=0 \end{cases}$ $f$ is Borel-measurable: Proof (of Daniel Wainfleet). It suffices to ...
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Finding Borel sets

Consider a function $f:\mathbb R\to\overline{\mathbb R}$ defined as $$f(x)=\begin{cases}\frac{1}{x}, x\neq 0\\ \infty, x=0 \end{cases}$$ Is $f$ Borel-measurable? I followed the answer given here ...
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Why can simple functions not take the value $\infty$?

When one develops the theory of integration, why is it usually the case that simple functions are not allowed to take the value $\infty$? Recall that if $(X,\mathcal{M})$ is a measureable space then ...
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1answer
17 views

Reweighting preserves positive average

Let $h:[0,\infty) \to \Bbb R$ measurable and $\int_0^\infty \vert h(y) \vert \text d y <\infty$ and suppose $$\int_0^c h(y) \text d y \geq 0 \quad \forall c >0$$ Let $\omega : [0,\infty ) \to [0,...
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Prove the following statement about measurable sets and functions

Let $(f_n)_{n = 1}^{\infty}$ be a sequence of measurable functions defined on a measurable set $E$, $m(E) < \infty.$ Suppose that $(\forall x \in E)(\exists M_x \in \mathbb{R})$ such that $\sup_n \...
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1answer
42 views

Is $f$ measurable function?

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $$f(x) = \left\{\begin{matrix} \ln(e+x), & x\in\mathbb{N}\\ x^3, & x\in(-\infty, 0]\\ e^{-x}, & x\in(0,\infty)\setminus\mathbb{...
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Borel-measurable function

The original question is: Let $(\Omega, \mathcal{F})$ a measurable space with $\mathcal{F}=({\emptyset , \Omega})$. Show that a function $f:\Omega\to\mathbb{R}$ is borel-measurable, if and only if $f$...
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63 views

Lebesgue Integral equal to Lebesgue Integral over partitions

I did the proof though I didn't use the hypothesis of $X=E\cup F$. Also I didn't see the why of $f$ to be in $\overline{\mathbb R}.$ Could this be any $Y$ space? Let $f:X\to\overline{\mathbb R}$ be ...
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1answer
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Time dependent process and progressively measurability

I'm trying to show that $f : \left[0,\infty\right)\times\mathbb{R}\rightarrow\mathbb{R}\text{ is a Borel function},X : \Omega\times\left[0,\infty\right)\text{ is a progressively measurable process}\...
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Measurability of closed opeartor

Given separable Banach spaces $(E, \Vert \cdot \Vert_E)$ and $(F, \Vert \cdot \Vert_F)$. The Banach space $E$ is endowed with sigma algebra $\mathcal{F}$ which is generated by the open set of it. ...
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“if $f:R \to R $ is $\mathbb{A}$ -measurable then for all $\delta>0$ exist $a \in \mathbb{R} $ such that $(f^{-1}(a,a+\delta))^c $ is countable ”

Take $X \neq \emptyset$. Let $\mathbb{A}$ be the following $\sigma-$algebra: $$\mathbb{A}=\{ A \in P(X) : A \text{ is countable or } A^c \text{ is countable} \}$$ I want to know if it's true or ...
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How to go about finding all $\mathcal{F}-$measurable functions

Let $\Omega$ be uncountable and $\mathcal{F}:=\{A \subseteq \Omega: A \operatorname{or} A^{c} \operatorname{countable}\}$ Find: All $f: \Omega \to \mathbb R$ that are $\mathcal{F}-$measurable. With ...
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Prove that $g(x):=\int_0^1f(x,y)dy$ is Borel measurable.

Let $f: [0,1]\times [0,1]\to\mathbb R$ satisfy: (i) for each $x\in [0,1]$, the function $y\mapsto f(x,y)$ is Riemann integrable on $[0,1]$; and (ii) for each $y\in [0,1]$, the function $x\...
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How to map the coefficients of an SDE to its solution process in a measurable way?

Preliminaries and Standard Technical Framework Let $T \in (0, \infty)$ be fixed. Let $d \in \mathbb{N}_{\geq 1}$ be fixed. Let $$(\Omega, \mathcal{G}, (\mathcal{G}_t)_{t \in [0,T]}, \mathbb{P})$$ ...
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Stochastic process with measurable set of indices

Let $\zeta_t$ be a random variable from probability space $(\Omega, \mathcal F, P)$ to some measurable space $(X, \mathcal X)$. Assume that indices set $T$ for process $\zeta_t$ is also a measurable ...
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Is a function on a product measurable space measurable iff it is componentwise measurable?

Let $(\Omega_1, \mathcal{G}_1), (\Omega_2, \mathcal{G}_2), (\Omega, \mathcal{G})$ be measurable spaces and let $$(\Omega_1 \times \Omega_2, \mathcal{G}_1 \otimes \mathcal{G_2}) $$ the canonical ...
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Prove $φ_n =2^{-1+2n}(-1+2^n)\to f$ pointwise

(Particular case of the general theorem and proof) Let $f:X\to[0,\infty]$ be measurable function defined on the interval $(k2^{-n},(k+1)2^{-n}]$ and $φ_n:X\to [0,\infty)$ monotone increasing ...
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Rigorous construction of the pointwise limit of a sequence of random variables

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space and let $$X_1,X_2,X_3,... \: \Omega \rightarrow \mathbb{R} $$ be a sequence of random variables. Moreover, let there be an event $A \...
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Does taking the limit in one argument preserve regularity in another argument?

Preliminaries Let $T>0$ be a positive constant, $d,n \in \mathbb{N}_{\geq 1}$ be natural numbers and $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space. For $m \in \mathbb{N}_{\geq 0}$, ...
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Notation recommendation for coarsening a probability distribution

Assume some finite support probability measure $\mu$ on a sigma algebra $\Sigma(X)$. Now in some cases it becomes necessary to coarsen the underlying sigma algebra according to the following method: ...
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Sequence of nonnegative measurable functions which decreases to a function f

Let $(f_n)$ be a sequence of nonnegative measurable functions which decrease pointwise to a function $f$ on a measurable set $E$, and suppose that $\int_E f_k < +\infty$ for some $k$. Prove that $\...
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1answer
33 views

Approximate the integral of an unsigned measurable function

I've been struggling with this problem. Let $f\colon X \to [0,+\infty]$ be an unsigned measurable function. Suppose $\int f < \infty$. Prove that for every $\epsilon > 0$ there exists a set $E ...
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1answer
14 views

Showing that transition is measurable

Let $P:\mathbb{R}^n\times \mathscr{B}_{\mathbb{R}^n} \rightarrow [0,\infty]$ be a function such that $P(x,-)$ is a probability measure for each $x$ and $P(-,A)$ is (borel) measurable for each $A$. ...
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Does piecewise continuous imply Borel measurable?

It's extremely well-known that continuous functions are Borel measurable, but what about piecewise continuous functions? For the Lebesgue measure, I suspect that we'd have a proof as simple as "...
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How to show this function is measurable

This is a problem from the S.-T YAU COLLEGE MATH CONTEST in 2012. The original problem is Let $f(x)$ be a real measurable function defined on $[a,b]$. Let $n(y)$ be the number of solutions of the ...
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σ-algebra and measure of the union of 2 sets. Based on measure theory.

A measure space $(\mathbb S,\mathcal S,μ )$ is not complete. The system of all its null sets is $\mathcal O$. Let $$\mathcal S' = \{A ∪ O : A ∈ \mathcal S, O ∈ \mathcal O\}.$$ The formula of the ...
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Show that $f_n$ converges in $L^1$ norm

Suppose that $f_n: X \to C$ are a dominated sequence of measurable functions, and let $f:X \to C$ be another measurable function. Show that $f_n$ converges in $L^1$ norm to $f$ if and only if $f_n$ ...
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1answer
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How to express random inequality events in terms of measurable events?

Suppose that $(\Omega,\mathcal{F},P)$ is a probability space, and $X,Y:(\Omega,\mathcal{F}) \rightarrow (\mathbb{R},\mathcal{B}(\mathbb{R}))$ are real random variables. I want to justify why an event ...
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suppose $ f: \mathbb{R} \rightarrow \mathbb{R}$ is M-measurable, Let E be the set of points at which f is continuous, show that E is M-measurable.

I know that if f is M-measurable, that for any open set U, $f^{-1} (U)$ is M-measurable, so I tried to show the value of the set is union of open sets but then I realized that is not true for example $...
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Correct Formulation of a map between two measurable spaces

Let $\pi: (X,\mathcal{M},\nu) \to (Y,\mathcal{N},\eta)$ be a measurable map i.e. $\pi^{-1}(E) \in \mathcal{M}$ for all $E \in \mathcal{N}$. I want to define a map from $L^{\infty}(Y,\eta)$ to $L^{\...
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Is the function $\displaystyle\liminf_{t\to t_0}f_t(x)$ Lebesgue measurable?

Let $\{f_t:[0,1]\to \mathbb R \ |t\in I\}$, where $I\subset \mathbb R$ is an interval, be a class of Lebesgue measurable functions. Then is it true that the function $\displaystyle\liminf_{t\to t_0}...
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1answer
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Is the set of Borel measurable transformations the closure of continuous functions under pointwise limits?

If $S$ and $T$ are topological spaces, then $f:S\rightarrow T$ is called a Borel-measurable transformation if for every Borel set $B$ in $T$, $f^{-1}(B)$ is a Borel set in $S$. My question is, is the ...
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1answer
196 views

When does a subset of a Polish space meet all the orbits?

While I was studying Borel actions of Polish groups on Polish spaces (I assume of measure $1$), I have tried to understand if a measure $1$ (hence dense) subset of this Polish space meets all the ...
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Measurable functions.

Let $f(x,y)$ be nonnegative Borel function on $\mathbb{R^2}$.Prove that $\psi(x) = \int_{\mathbb{R}}f(x,y)d\mu(y) $ is also Borel function($\mu $ is symbol for Lebesgue measure). My attempt: () $f(x,y)...
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How to Modify a Borel function in a Borel way-Self study

I have the following questions: assume that $X$ be a standard Borel space (i.e. a Polish space equipped with the $\sigma$-algebra generated by open sets) with a (possibly Borel, invariant, ergodic) ...
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1answer
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Measurability and induced probability distribution of an uncountable family of random variables

I've been trying to figure this out for a long time and I cannot seem to wrap my head around it: a) We have an index set $\mathcal{I}=[0,1]$, and for each $i\in\mathcal{I}$, $x_i$ is drawn i.i.d. ...
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1answer
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Basic Question on Conditional Expectation being in $L^2$

Suppose that $X,Y$ is a random-variables in $L^2(\Omega,\mathcal{F};\mathbb{P})$, where $(\Omega,\mathcal{F};\mathbb{P})$ is a complete probability space. Then the conditional expectation $E[X|\sigma(...
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Show that $f$ is $\mathbf{X}$-measurable iff $f^{-1}(E)\in\mathbf{X}$ for every Borel set $E$.

I'm trying to do exercises $2.N.$, $2.O.$, and $2.P.$ from Chapter $2$ of Bartle's The Elements of Integration and Lebesgue Measure.            &...
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1answer
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Doubt in understanding theorem 4.1 Real analysis Stein and Shakarachi

I came across the following theorem. I understood that $F_k(x)\to f(x)$ pointwise as $k\to \infty$, but I do not understand how $E_{l,j} $ is defined over a range of $F_k$. I am not able to visualise....
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1answer
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Approximating a bounded measurable function from below by a sequence of smooth functions

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a bounded measurable function. Is it possible to find a sequence of functions $\{f_n \}_n: \mathbb{R} \to \mathbb{R}$ in $C^{\infty}_c( \mathbb{R})$ ...
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Problem seems too easy. Is there a trap? Show that $f(1-f)=0$ almost everywhere…

So this is a problem I found on a qualifying exam and it has been nagging at me for some a while. It seems overly easy. Assume that $f$ is a nonnegative function on $[0,1]$ and \begin{align} \int_{[...
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Direct sum of subgroups of $\mathbb{Q}^n$ with $\mathbb{Q}$ is a Borel map - Self study

Let $\mathcal{Pow}(\mathbb{Q}^n)$ be the power set of $\mathbb{Q}^n$ and consider the product topology induced by the natural bijection $\mathcal{Pow}(\mathbb{Q}^n)\cong 2^{\mathbb{Q}^n}$ defined by $...
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2answers
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Proving $\omega\mapsto B(\cdot, \omega)$ is measurable.

Let $(B_t)_{t \geq 0}$ be a Brownian motion on the probability space $(\Omega, \mathcal{A},P)$, and $\pi_t$ the canonical projection at time $t$, and $C(\mathbb{R}^+_0,\mathbb{R}^d)$ is the set of ...
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21 views

Question on measurable functions being approximated by step functions

This question is based on Theorem 4.3 in Stein's book. It's trying to show that $f = \chi_{E}$, where $\chi_{E}$ is the characteristic function on a measurable set $E$, can be approximated by step ...
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1answer
44 views

Almost everywhere bounded function

Let $\psi$ be measurable function such that for all $ f \in L^{1}(\mu)$, $ f\psi \in L^{1}(\mu)$. Then $\psi$ is bounded almost everywhere(in sense of measure $\mu$). My attempt: Use contraposition: ...
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1answer
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A consequence of the Selection Theorem for the Effros Borel space F(X) - self study

In Kechris' textbook "Classical Descriptive Set Theory", the following exercise is stated (pp. $77$, Exercise $(12.14)$): "Let X be a measurable space and Y a Polish space. Show that a function $f\...
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$\mathbb{E}[f(X,Y)|Y]=\mathbb{E}[f(X,Y)]$ for all bounded measurable $f$ using the monotone class theorem

Let $X,Y$ be independent rv's. For $f:\mathbb{R}^2\to\mathbb{R}$ measurable we have $f_X:\mathbb{R}\to\mathbb{R}$ where $$f_X(y)= \left\{ \begin{array}{lr} \mathbb{E}[f(X,y)] & : \...