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Questions tagged [measurable-functions]

For questions about 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 ...
mathnoob's user avatar
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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 \begin{equation} f: H^n \to H^n \\ A = \sum_i \lambda_i e_i e_i^T \mapsto \sum_i f(\lambda_i) ...
2000mg Haigo 's user avatar
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If $f_n \rightarrow f$ almost everywhere and all $f_n$ are measurable, how can I redefine $f$ on a null set to make it measurable?

I am self-studying real analysis out of Folland, and I am confused at the beginning of his proof of the Dominated Convergence Theorem. It reads as follows, where all $f_n$ map from measure space $(X,M,...
Lightbulb's user avatar
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if $f_n \uparrow f$ prove $\mu(f_n\geq t)\rightarrow \mu(f\geq t)$

I think I have to use montone convergence theorem and I have followed this line the sets $f_n\geq t$ are increasing $\lim\mu(f_n\geq t)=\mu(\cup(f_n\geq t)=?\mu(f\geq t)$ if $x\in \cup(f_n\geq t) \...
Dsrksidemath's user avatar
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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 $$f(t) := \int_\Omega \varphi(x)u(t,x) \mathrm{d}x.$$ ...
C_Al's user avatar
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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 ...
HIH's user avatar
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Measurability of $\int_\Omega u(t,x)\;dx$ given $t \mapsto u(t,x)$ is measurable

Let $\Omega$ be a bounded domain and consider a function $u\colon [0,T] \times \Omega \to \mathbb{R}$. Suppose that $$F(t) := \int_\Omega u(t,x)\;dx$$ exists for all $t \in [0,T]$. If I know that $t \...
C_Al's user avatar
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equivalence for Borel-measurable function

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be the function $f\left( x \right)=\left| x \right|$ and $\mathcal{A}= f^{-1}\left( \mathcal{B_1} \right)$ denote the preimage $\sigma$- algebra of $f$. ...
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BV function $f$ is $\|Df\|$-measurable

Let $f:\mathbb R\to\mathbb R$ be a $L^1$-function of bounded variation where the total variation measure $\|Df\|(\mathbb R)$ is bounded, i.e. $f\in BV(\mathbb R)$. Since $f\in L^1(\mathbb R)$, we have ...
MATH's user avatar
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Does any measurable function whose integral over any interval is $0$ satisfies $f(x)=0, a.e.$? [duplicate]

If a (Lebesgue) measurable function $f: ℝ→ℝ$ satisfies $∫_a^b f(x)dx = 0$ for any $a, b ∈ ℝ$, does $f(x)=0, a.e.$? I understand that $f(x)=0$ follows if $f$ is continuous, but I was not sure if the ...
Jaborandi Kakapo's user avatar
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Le Gall Exercise 1.3

I am trying to do exercise 1.3 from Le Gall's Measure theory, probability and stochastic processes: Let $C([0,1],\mathbb{R}^d)$ be the space of all continuous functions from $[0,1]$ into $\mathbb{R}^d$...
Ishigami's user avatar
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Image of measurable sets under one to one (a.e) functions

I want to know about this question that is image of a measurable set under a one to one (almost everywhere) function, measurable? Consider the following: Let $(X, \mathcal{B}_X)$ and $(Y, \mathcal{B}...
Reza Yaghmaeian's user avatar
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A problem of measurability of stochastic kernel

I want to use the stochastic kernel theorem to prove the existence of the following measure: I have a probability space $(\Omega,\mathcal{F},P)$ and and infinite-dimensional measure space $(A,\...
houssem agili's user avatar
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Gaussian measure measurability

I want to use the transition probability theory to prove the existence of a measure on a product space : I have two spaces $(\Omega,\mathcal{F},P)$ and for each $\omega \in \Omega$ i have another ...
Houssem Ajili's user avatar
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A conjecture about measurable functions [duplicate]

(Modified to be specific enough so that the other post does not answer my question) I have the following conjecture about measurable functions. Let $\Omega$ be the sample space equipped with $\sigma$-...
vietajumping's user avatar
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Null function and Lebesgue measure on $\mathbb{R}^N$

We know that some measurable function $f:(0,T)\times\Omega\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ has the following property: For almost all $t\in (0,T)$ (Lebesgue measure in $\mathbb{R}$) ...
Bogdan's user avatar
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When is measurability of a function $f$ equivalent to $f$ being almost everywhere the limit of measurable functions?

I have a suspicion that the following is true: $\def\AAA {\mathcal{A}} \def\BBB {\mathcal{B}} \def\NN {\mathbb{N}}$ Theorem: let $(X,\AAA,\mu)$ be a complete measure space, and $(Y,\BBB)$ a measurable ...
Sam's user avatar
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2 answers
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If $\int_Afd\mu\geq0$ for all $A\in\mathscr{A}$, then $\int f\chi_Ad\mu=0$ for $A=\{x\in X:f(x)<0\}$

I am self-studying measure theory using Measure Theory by Donald Cohn. I am confused by his proof of the following result: Corollary 2.3.13$\quad$ Let $(X,\mathscr{A},\mu)$ be a measure space, and ...
Beerus's user avatar
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Monotone functions on $\mathbb{R}^n$

I am looking for references on the following topic. It is a known fact that, every measurable function $f:\mathbb{R} \to \mathbb{R}$ monotone can be seen as the sum $f = f_c + f_j$ where $f_c$ is a ...
Nestor Bravo's user avatar
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1 answer
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Question About Function Integrability - Proposition 2.3.10 from Measury Theory by Donald Cohn

I am self-studying measure theory using Measure Theory by Donald Cohn. The book makes the following definition: Definition$\quad$ Suppose that $f:X\to[-\infty,+\infty]$ is $\mathscr{A}$-measurable ...
Beerus's user avatar
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Proof of $\int_{\Omega_2}f(y)\mu{\circ}T^{-1}(dy)=\int_{\Omega_1}f(T(x))\mu(dx)$

Theorem. Suppose $(\Omega_1,\mathcal A,\mu)$ is a measure space and $(\Omega_2,\mathcal A_2)$ is a measurable space. If $T:\Omega_1\rightarrow\Omega_2$ and $f:\Omega_2\rightarrow\bar{\mathbb {R}}$ are ...
reyna's user avatar
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2 votes
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Question on the Construction of the Integral in Measure Theory

I am self-studying measure theory, and I got some trouble understanding the construction of the integral. Here is the first two stages of the construction: Stage 1$\quad$ We begin with the simple ...
Beerus's user avatar
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Joint measurability of strongly continuous vector-valued function

Suppose that $\mathbb{R}\to L^2(\mathbb{R}),$ $t\mapsto g_t$ is continuous. (Under which additional assumptions) is it true that $G\colon \mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$, $G(t,x):=\...
ym94's user avatar
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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 ...
Krystal Justin's user avatar
3 votes
0 answers
61 views

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)....
Sarvesh Ravichandran Iyer's user avatar
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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 ...
Perelman's user avatar
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1 answer
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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 ...
jay sri krishna's user avatar
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Random measure and empirical spectral distribution

I have a question about the application of the definition of a random measure on the empirical spectral distribution. Let $(X , \mathcal{B})$ be some measurable space and let $(\Omega, \mathcal{F}, \...
MathAccount12's user avatar
2 votes
0 answers
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Measurability of an integral operator?!

Is it possible to prove the measurability of the following map $\Phi_n \colon (C(\mathbb{R}^d), \sigma(\mathcal{C})) \to (\mathbb{R}, \mathcal{B}(\mathbb{R}))$, $n \in \mathbb{N}$, defined by $$ \...
TrippyMushroom95's user avatar
2 votes
1 answer
91 views

Show that the measure $\mu ([a,b]) = \ln \left( \frac{1+b}{1+a} \right)$ is invariant for function $\Phi_k$.

A measure is defined as: $\mu ([a,b]) = \int_a^b \frac{1}{1+x} = ln \left( \frac{1+b}{1+a} \right)$ The function $\Phi_k$ is defined as: $\Phi_k = kx$ for: $x \in \left(0, \frac{1}{k} \right)$ $\...
thefool's user avatar
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3 votes
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Is the median a measurable function of the probability distribution?

For $\mu \in \mathcal P(\mathbb R)$, let $m(\mu)$ be the median of $\mu$, defined as the smallest of all medians of $\mu$ as follows: $$ m(\mu) = \inf \left\{ x \in \mathbb R \,\middle|\, \mu((-\infty,...
Cyril B.'s user avatar
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1 answer
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Show the existence of a sequence of simple functions such that $\lim_{n\to\infty}\varphi_n(x)=f(x)$ and $\Vert\varphi _n(x)\Vert\leq\Vert f(x)\Vert$

Let $(X,\Sigma)$ be a measurable space and $f:X\to \mathbb{R}^d$ be a measurable function. Suppose that $\Vert \cdot \Vert $ is a norm of $\mathbb{R}^d$. How can I show that there's a sequence $(\...
rfloc's user avatar
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1 vote
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Definition of a measurable function with respect to $\sigma$ - algebra

A function $X:(\Omega , \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{B}) $ is called a measurable function(or random variable with probability $P$) if $X^{-1} (B)= \{ \omega \in \Omega : X(\omega) \...
JAEMTO's user avatar
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Continuous process is measurable

I am revising the lecture notes and i stumbled across this proposition: Given $(\Omega, \mathcal{A}, P)$ a probability space, the following holds: If a process $(X_t)_{t\geq 0}$ has continuous ...
Davide's user avatar
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1 answer
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measurability of a random variable taking values in the space of continuous functions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability sapce and let $X: \Omega \to C^+(\mathbb{R})$ be a random variable taking values in the space of real valued non-negative continuous functions ...
mathematico's user avatar
1 vote
1 answer
48 views

Is it true: $\int ( \int f(u,z) d\mu(u))^2 dz \leq \|\mu \|_{TV} \int ( \int f(u,z) du)^2 dz$?

I am trying to understand if for a measurable continuous nonnegative function $f$ on $\mathbb{R}^2$, $\sigma$-finite measure $\mu$ on $\mathbb{R}$ such that $\mu(A)>0$ for any measurable set on the ...
Grandes Jorasses's user avatar
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1 answer
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Nice application of dominated convergence theorem

Let $\delta \in \mathbb{R}$, $$f(x)=\frac{sin(x^2)}{x}+\frac{\delta x}{1+x}.$$ Show that $$\operatorname{lim_{n\to \infty}} \int_{0}^{a}f(nx)=a\delta$$ for each $a>0.$ I am unable to find ...
Infinity's user avatar
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4 votes
0 answers
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Limit of measurable functions in a general metric space is measurable?

Let $(f_n)_{n \geq 1}$ be a sequence of measurable functions from $(\Omega, \mathcal F)$ to a metric space $(X, d)$. Suppose that $f$ is such that $$ \lim_{n \rightarrow +\infty} d(f_n(\omega),f(\...
Jeffrey Jao's user avatar
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1 answer
68 views

Graph of a measurable function is measurable (not on metric spaces)

Im trying to solve the following problem: Problem Let $(S, \mathcal{A}) , (S, \mathcal{A} ')$ be measurable spaces and $\varphi, \psi , \psi' : S \to S' $ $\mathcal{A}- \mathcal{A}'$ measurable ...
DeeJeiK's user avatar
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0 answers
39 views

Decompose measurable function into characteristic functions [duplicate]

Let $(X, \mathcal{A},\mu)$ be a measure space and $f : X → [0,∞]$ be measurable. Show that there exist measurable sets $A_k, k = 1,2,...$ such that $$f = \sum_{k = 1}^{\infty} \frac{1}{k} \chi_{A_k}$$ ...
The Limit Does Not Exist's user avatar
2 votes
1 answer
105 views

Understanding a measurabiliy statement from Section 6.3 in Lehmann and Romano

This paragraph is at the end of Section 6.3 in the book Testing Statistical Hypotheses by Lehmann and Romano: In most applications, $M(x)$ is a measurable function taking on values in a Euclidean ...
Alphie's user avatar
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0 answers
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Construct a non Lebesgue measurable function with given condition. [duplicate]

Construct a function $f : \mathbb{R} → \mathbb{R}$, which is not Lebesgue measurable, but $f^{−1}(\{x\})$ is Lebesgue measurable for every $x \in \mathbb{R}$.
The Limit Does Not Exist's user avatar
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1 answer
44 views

Why is $1_{\{\omega : X(\omega) \geqslant t\}}(\omega) = 1_{\{t: t\leqslant X(\omega)\}}(t)$.

I cannot see the logic behind how $1_{\{\omega : X(\omega) \geqslant t\}}(\omega) = 1_{\{t: t\leqslant X(\omega)\}}(t)$ in the proof here, Expectation of $\mathbb{E}(X^{k+1})$ Could someone please ...
Trajan's user avatar
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1 vote
0 answers
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Question on how to approach measurability questions for real valued functions

My question is regarding the following Proposition: Let $(X,\mathcal{A})$ be a measurable space and let $f: X \rightarrow \mathbb{R}$ be an $\mathcal{A}$-measurable function. If the set $C$ is in the ...
Maxi's user avatar
  • 97
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0 answers
14 views

Existence of measurable maximizers

Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha ...
FeleMath's user avatar
0 votes
1 answer
50 views

Question regarding the definition of measurable functions

Consider the following definiton: Definition: Let $(X, \mathcal{A})$ be a measurable space. A function $f:X \rightarrow \mathbb{R}$ is measurable if {$x \in X:f(x)>a $}$ \in \mathcal{A}$ for all $a ...
Maxi's user avatar
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0 answers
19 views

Measurability of the Wiener measure with respect to the starting point

Let $M$ be a Riemannian manifold, and let $C = \{ c : [0,1] \to M \mid c \text{ is continuous}\}$. Endow $C$ with the Wiener measure $\mathbb P_x$ concentrated on the curves $c \in C$ with $c(0) = x \...
Alex M.'s user avatar
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1 vote
0 answers
75 views

A bounded measurable function

Let $g:R→R$ be a non-negative integrable function. Let $f:R→R$ be a bounded measurable function satisfying $f(x)>1$ for every $x∈R$. Suppose that $∫_R f^n g≤M$ for every $n∈N$. Show that $g(x)=0$ ...
user1281744's user avatar
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0 answers
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Oksendal. Stohastic differential equations. Problem 4.7

I am stacked at problem 4.7 (b) in the book Oksendal. Stohastic Differential equations. The conditions of the problem are the next: Let $X_{t}$ be an Ito integral. $dX_{t}=v\left(t,\omega\right)dB_{t}\...
Grigogiy Reznichenko's user avatar
5 votes
1 answer
223 views

Identically distributed random variables and events of probability $0$

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $X_1,\dots,X_{n+1}:\Omega\to \mathbb{R}$ be random variables. Suppose that the random variables are identically distributed, i.e. $$\...
S.H.W's user avatar
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