Questions tagged [measurable-functions]

For questions about measurable functions.

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Neyman-Pearson Infimum Integral non rand. Test

*Show for 2 distributions $P_0$ and $P_1$ on $(X,F)$ with densities $f_0$ and $f_1$ with respect to $\mu$ that $inf\{P_0(\phi=1)+P_1(\phi=0) | \phi: X \rightarrow\{0,1\} \text{ non rand. test}\}$ =$\...
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$\sigma$-algebra generated by uncountable random variables

Let $(X_1,\mathcal{A}_1)$, $(X_2,\mathcal{A}_2)$, and $(X_3,\mathcal{A}_3)$ denote three measurable spaces. Let $\mathcal{G}$ denote a set of (pontentially uncountable) measurable functions $g:X_2\to ...
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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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Question on merging Expected values

For my course on financial mathematics i have the following exercise (with solution): What I do not get about this solution is how they get $\xi$ to turn up everywhere. For example, the merge $E[Y|G] ...
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limit involving measure [closed]

let $f$ be a measurable function of $\mathbb R^N$ with limited support and not integrable. Define $F_n=(x \in \mathbb R^N| n^2<|f(x)| \le (n+1)^2)$. Prove that $\limsup_{n\rightarrow +\infty }(n^...
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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 ...
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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 ...
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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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Question involving integration of nonnegative measurable functions using simple functions

Question: I am working on a problem in Folland's Real Analysis (specifically, 2.16) and I want to justify something: Suppose $f$ is a measurable nonnegative function and $\int f<\infty$. Let $\...
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Is the lower Hausdorff dimension of measures a Borel measurable function on measure space?

Let $X\subset\mathbb{R}^n$ be a compact subset, $\mathcal{B}(X)$ be the Borel $\sigma$-algebra on $X$. Denote $\mathcal{M}(X)$ the collection of all Borel probability measures on $X$. It is clear that ...
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Can a sequence of non-eventually-$G$-measurable random variables var-converge to a $G$-measurable random variable?

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $G$ be a sub-$\sigma$-field of $\mathcal{F}$. Suppose that $(X_k)_{k \in \mathbb{N}}$ is a sequence of square-integrable real-...
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If this sigma algebra contains singletons

Let $\Omega$ be a compact Hausdorff space in $\mathbb{C}^n$. Let $\sigma_\Omega$ be the Borel sigma algebra on $\Omega$. Let $\zeta: \Omega\longrightarrow\partial \mathbb{D}$ be a non constant ...
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If $f(x)$ is measurable function for $x\in\mathbb{R}$, then if $h(x,t)=f(x+t)$ also measurable in $<x,t>\in\mathbb{R}^2$

So for $h(x,t)$ to be measurable it means for any $a\in\mathbb{R}$ that $Z(a)=\left\{<x,t>\big|h(x,t)\geq a\right\}$ is a measurable set Since $f(x)$ is measurable, then we have $E(a)=\left\{y\...
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Question about functions in $L^2$ space

Let $f,g \in L^2 (\mathbb{R})$. For $n \in \mathbb{N}$, define $f_n \in L^2(\mathbb{R})$ as $f_n(x) = f(x-n)$ for all $x \in \mathbb{R}$. Prove that $\lim_{n \rightarrow \infty} \int_{\mathbb{R}} f_n ...
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Show that $\phi_n(x) \leq \phi_{n+1}(x)$ (Folland Theorem 2.10)

I am reading the proof of Theorem 2.10 in Folland's Real Analysis. I'm stuck to show the sentence: It is easily checked that $$\phi_n \leq \phi_{n+1}$$ for all $n$. I first noticed that $E_{n}^{k}=...
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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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2 answers
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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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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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Relationship between Disintegration Theorem and Co-Area formula

I have a feeling the Disintegration Theorem and the Co-Area formula for Lipschitz functions are actually very much related but I cannot seem to formalize it. DISINTEGRATION THEOREM: Let $(\mathsf{X}, ...
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Is it enough to check ergodicity for intervals? [closed]

Let $T:[0,1]\to[0,1]$ be a Lebesgue measurable map which preserves the Lebesgue measure (indicated here by $m(\cdot)$). Suppose that $m(T(I))= 1$ for every nonempty $T$-invariant interval $I\subset [0,...
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Convergence in distribution and almost sure convergence

Let $(\Omega, \mathcal{F}, \mathbb{P})$ a probability space and $Z_n, Z$ defined on it and taking values in $\mathbb{R}$, such that $Z_n \to Z$, $\mathbb{P}$-almost surely. Let $g_n, g \colon \mathbb{...
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2 votes
1 answer
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How to compute $\lim_{u\to u_{0}^{-}} \mu\left( f^{-1}(u,u_{0}] \right)$?

If $f\colon I\subseteq\mathbb{R} \rightarrow \mathbb{R}$ is a Lebesgue-measurable function ($I$ is a closed interval), $\mu$ is the Lebesgue measure, my question is whether the limit $$\lim_{u\to u_{0}...
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1 answer
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Inverse image of a half line is an event if and only if the inverse image of all Borel sets is an event.

Definition 1. Let $(\Omega,\mathcal{F}, P)$ be a probability space. A random variable is a function $X=X(\omega): \Omega \rightarrow \mathbb{R}$ such that, for any $B \in \mathcal{B}(\mathbb{R})$, $$ ...
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1 vote
1 answer
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The existence of a set with nonzero measure on which two measurable functions $f>g$ is "separated" by a constant.

Suppose $(\Omega,\mathscr{F},\mu)$ is a measure space and $f,g$ are $(\mathscr{F},\mathscr{B})-$measurable functions where $\mathscr{B}$ is the Borel algebra on $\mathbb{R}$. If $$ \mu(\{f<g\})>...
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1 answer
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Essential sup Property

Let $(X,\mathfrak M, \mu)$ be a measure space, i.e. $\mathfrak M$ is a $\sigma$-algebra over the set $X$ and $\mu: \mathfrak{M}\to [0,+\infty]$ is a measure. Let $f:X\to \overline{\mathbb R}:=[-\infty,...
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7 votes
1 answer
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Is $f:[a,b] \times \Omega \to E$ measurable?

Problem: Suppose that $[a,b] \subset \mathbb{R}$, $(\Omega, \mathcal{F})$ is a measure space and $E$ is a topological space. Suppose $f : [a,b] \times \Omega \to E$ is such that: $\forall t$ $f(t, \...
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1 answer
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Showing that if $F\subset X,F\in Ba(X)$ is closed, $A\subseteq F$ implies $A\in Ba(F)\iff A\in Ba(X)$ - important for Baire regularity

$\newcommand{\ba}{\mathcal{Ba}}$I understand that there are different used definitions, so here are mine: The Baire $\sigma$-algebra $\ba(X)$ on a topological space, $X$, is the smallest $\sigma$-...
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auxiliary statement for the central limit theorem

I can't show this statement: P(X ≤ x_0) ≤ E[f(X − x_0)] ≤ P(X ≤ x_0 + ε). X is a random Variable. f : R → [0, 1] and f is a measurable function with the properties: f(x) = 1 for x ≤ 0, f(x) = 0 for x ≥...
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regarding lebesgue integral

as in the wikipedia page, we define the Lebesgue integralon non-negative functions: Let f be a non-negative measurable function on E, which we allow to attain the value +∞, in other words, $f$ takes ...
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the relationship between strongly measurable and measurable.

In real analysis, we know Consider any mapping $f: X \rightarrow Y$. If $(X, \mathcal{N})$ and $(Y, \mathcal{N})$ are measurable spaces, a mapping $f: X \rightarrow Y$ is called $(\mathcal{M}, \...
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$f$ is $\mathbb{B}$ measurable if and only if $g$ is $\mathbb{B}$ measurable and $P,N \in \mathbb{B}$

Let $\mathbb{B}$ be a $\sigma$-algebra on a set $X$ and let $f\,:\,X \rightarrow \overline R$ be an extended real-valued function. Define the sets $P\:= f^{-1}(-\infty)$ and $N\:=f^{-1}(+\infty)$. ...
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2 votes
1 answer
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Confusion in showing that $f$ is measurable

Question: Show that \begin{align} f(x) = \begin{cases} -3x, &x < 0,\\ x, &0 \leq x \leq 2\\ 1 - x, &x > 2 \end{cases} \end{align} mapping $(\mathbb R, \mathcal B_{\mathbb R}, \mu_{\...
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3 votes
1 answer
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Are Borel measurable functions closed in pointwise topology?

Let $X$ be a metrizable space. The Lebesgue-Hausdorff theorem states that the minimal class $\mathcal{C}$ of functions $f :X \to \mathbb{R}$ closed under pointwise limits of sequences, such that $C(X) ...
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3 votes
2 answers
277 views

Measurability conditions in the definition of Brownian motion

I am trying to study stochastic integration from Kuo, Introduction to stochastic integration. I have two small questions about the definition of Brownian motion. First question: Kuo presents it as a ...
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1 vote
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Is $1/(T-t)$ an almost everywhere finite function for $t\in[0,\infty)$

Context: I'm reading theorem which requires me to check the following conditions: Let the vector valued function $f(t,x)$ be defined almost everywhere and measurable in the domain $G$ of the $(t,x)$-...
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2 votes
0 answers
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Integrability of the inner Hausdorff measure of noncompactness

Recall that the inner Hausdorff measure of noncompactness in a Banach space $Y$ is defined as $$ \chi_{i}(B):=\inf\{\varepsilon>0: B\subset \cup_{i=1}^{n} B(y_{i},\varepsilon), \textrm{for some } ...
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3 votes
1 answer
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Showing that the preimage of the projection mappings $E_1\times\cdots\times E_n\mapsto E_i, E_1,\dots,E_n\in B(R)$ is contained in $B(R^n)$

Preamble: The main proof that $\mathcal{B}(\mathbb{R}^n) = \sigma(C), C = \{E_1\times\cdots\times E_n\mid E_1,\dots,E_N \in \mathcal{B}(\mathbb{R})\}$ is given here: Is $\mathcal B(\mathbb R^n)$ ...
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Showing that $E(X) = \int_{\Omega}X(\omega)d\mathbb{P}(\omega) = \int_\mathbb{R}xd\mathbb{P}_X(x)$

Let $X$ be a real-valued random variable on the probability space $(\Omega, \mathscr{F}, \mathbb{P})$, so that X's expected value is $E(X) = \int_{\Omega}X(\omega)d\mathbb{P}(\omega)$. Suppose that $X$...
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Is there a generalization of the measure theoretic change-of-variables formula to $\mathbb{R}^n$?

In Bogachev's Measure Theory, he states the following: Suppose we are given two spaces $X,Y$ with $\sigma$-algebras $\mathcal{A}$ and $\mathcal{B}$ and an $(\mathcal{A},\mathcal{B})$-measurable ...
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extension of measurable function from dense subset

This is the same question posted in https://mathoverflow.net/questions/415237/extension-of-measurable-function-from-dense-subset. Let $M$ be a compact riemannian manifold equipped with a geodesic ...
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2 votes
1 answer
37 views

Measurability of minimum with respect to lexicographic order

Let $f_i:(X,\Sigma)\to\mathbb R^2$ be Borel measurable maps for $1\leq i\leq n$. Consider the map $f:(X,\Sigma)\to\mathbb R^2$ defined pointwise by $$f=\min_{1\leq i \leq n} f_i$$ where the minimum ...
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0 answers
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Is a function on a product space measurable if it is equal almost surely to a product measurable function?

My question is the following: Consider Borel spaces $(S, \mathcal{S})$ and $(T, \mathcal{T})$, where $\mathcal{S}$ and $\mathcal{T}$ are Borel $\sigma-$algebra of $S$ and $T$ respectively. Consider a ...
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0 votes
2 answers
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Image of a measurable function is not measurable

I am trying to find an example that shows the image of a measurable function is not measurable. Here is what I found: Let $X=\{0,1\}$ with $\sigma$-algebra $\{\emptyset, X\}$. Let $f:X \rightarrow X$ ...
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1 vote
1 answer
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Weak topology and weak convergenge in probability spaces

Let $X$ be a Polish space (metrizable, complete, separable) with $\mathcal{B}(X)$ its borel sigma algebra. Let us consider $\mathcal{P}(X)$ the space of probability measures on $\mathcal{B}(X)$. We ...
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4 votes
0 answers
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Why can we use MCT in the proof of Fubini's theorem (in Stein's book)?

In Stein's Real Analysis, he uses six steps to prove Fubini's theorem. First he uses $\mathcal F$ to denote the set of integrable funcyions satisfying the conclusions in the theorem: $f\in L^1(\...
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