Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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Defining a measure $\lambda$ in terms of a sequence of measures $(\mu_n)$

Below is a problem concerning a sequence of measures. My trouble arises when trying to prove the countable additivity of our measure $\lambda$, which is defined in terms of a sequence of measures. (I ...
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2answers
103 views

integral computed with respect to a sub-$\sigma$-algebra

Let $\mathcal M_0$ be a $\sigma$-algebra that is contained in a $\sigma$-algebra $\mathcal M$ of subsets of a set $X$, $\mu$ a measure on $\mathcal M$ and $\mu_0$ the restriction of $\mu$ to $\mathcal ...
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2answers
337 views

Iterated conditioning

I understand how this holds: $\mathbb{E}(\mathbb{E}(X|\mathit{G})|\mathit{H}) = \mathbb{E}(X|\mathit{H})$ where $\mathit{H} \subset \mathit{G} $. Then, how does this hold: ...
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1answer
123 views

A technical question about the Lebesgue measure

Let $U$ be an open set in $\mathbb{R}^2$. How to prove that the boundary of the CLOSURE of $U$ has Lebesgue measure 0 ? Thanks.
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2answers
305 views

Existence of minimal $\sigma$-algebra and transfinite induction

It is well-known that: Given a set $X$ and a collection $\cal S$ of subsets of $X$, there exists a $\sigma$-algebra $\cal B$ containing $\cal S$, such that $\cal B$ is the smallest ...
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1answer
200 views

$\sigma(\mathcal{A}) = $ the set of countable unions of countable intersections of elements or complements of elements of $\mathcal{A}$

Let $\mathcal{A} \subseteq \mathcal{P}(\Omega)$, $\Omega$ a set. Then isn't the set of call countable unions of countable intersections of elements or complements of elements of $\mathcal{A}$ equal ...
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1answer
230 views

Continuous functions agreeing almost everywhere

Let $\Omega$ be a (non-empty) subset of $\mathbf{R}^n$. If $f$ and $g$ are continuous real-valued functions agreeing almost everywhere on $\Omega$, do they agree on the whole of $\Omega$? I am not ...
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144 views

When does it make sense to say “the smallest measurable set containing $x\,$”?

We know for the Borel $\sigma$-algebra that each singleton set is measurable. I was working on the problem of proving that each infinite $\sigma$-algebra has uncountably many members. My solution went ...
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3answers
36 views

Change of variables with a square

Can someone help me understand this a bit better: $\int (x-y)^2 dx = \int(y-x)^2dx$ as $(y-x)^2 = (x-y)^2$. Now, if I make the change $z = x-y$ in the one on the LHS I get: $\int z^2 dz$ as $dz ...
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1answer
126 views

Trace for $L^\infty$ functions?

I'm considering the following problem. Let $s\in L^\infty ((0,T)\times K)$ for some compact $K\subset \mathbb{R^n}$ be given. Consider the Steklov average in time of $s$, i.e. for $h>0$ and ...
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27 views

Which conditions to put on a one-parameter family of measurable functions so that they are all defined in the same set?

While working on a certain problem (some details below, in case they are useful) I come up with a family of (Borel) measurable functions indexed by a parameter $t$ and defined only almost everywhere. ...
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1answer
57 views

Measurability of section functions

Let $(X,\mathcal{A})$ be a measure space and $K$ a compact metric space wiht the Borel $\sigma$-álgebra $\mathcal{B}$. Is it true that if $f:X\times K\rightarrow \mathbb{R}$ is ...
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1answer
81 views

Lebesgue density of a sum of RV's with Lebesgue density

Let $(X_i)_{i\in\mathbb{N}}$ be a real valued stationary sequence where each $X_i$ has the same Lebesgue density. Does \begin{align*} \sum_{i=1}^n X_i\end{align*} have a Lebesgue density, too? I ...
2
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1answer
79 views

Convergence almost everywhere?

Consider the following two statements about a random sequence $X_n$: (1) $X_n \stackrel{a.e.}{\rightarrow} X$. (2) $\mathrm{P}\{|X_n-X|>\epsilon, \ i.o.\} = 0, \ \forall \epsilon>0$. (a.e. ...
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0answers
241 views

Outer measure defined in terms of open intervals vs. closed intervals

This should be a simple question (it's been a long summer break....). I am reviewing some measure theory and noticed in my notes from a class that we defined the exterior measure in terms of the ...
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1answer
49 views

Measurable subset of a product space has measurable sections

Let $(F, \mathcal{F})$ and $(G, \mathcal{G})$ be measurable spaces. How can we show that if $E$ is measurable with respect to $\mathcal{F}\otimes \mathcal{G}$, then for every $x\in F$, the set $\{y\in ...
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1answer
386 views

Atomless probability measure

Assume that $m$ is an atomless probability measure on $\mathbb{R}^{d}$. Let $\left( X_{1},\ldots ,X_{d}\right) $ be a random vector with law $m$. Are the marginal cumulative distribution functions of ...
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170 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
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1answer
83 views

Quick question about $\sigma$-algebras.

I have a quick question concerning $\sigma$-algebras. If A is a collection of subsets of a set X and Y is the $\sigma$-algebra generated by A, then can I conclude that every element of Y is either (1) ...
3
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1answer
137 views

Balls going away from the origin

Suppose we have $k$ open balls $B_1,\dots,B_k$ in $\mathbb{R}^n$ centered at $0$ (their radii may be different) and $k$ vectors $v_1,\dots,v_k\in\mathbb{R}^n$. Is it true that $\mu\left(\cup_{i=1}^k ...
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111 views

Covering argument

In proving Harnak's inequality (I am referring to this article: "On Harnack’s Theorem for Elliptic Differential Equations"Communications on Pure and Applied Mathematics Volume 14, Issue 3 ), Moser ...
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1answer
182 views

Outer measure discontinuous from below

I was trying to find an example of an outer Measure which is not continuous from below. These are the definitions I use An outer measure on $X$ is a function $\mu^\ast: \mathcal{P}(X)\to ...
4
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1answer
173 views

measures on the family of locally measurable subsets

I am contemplating over the exercise about the ways to extend a measure to a collection of locally measurable subsets. To be precise: Let $(X,\mathcal M,\mu)$ be a measure space. Assume that $\mu$ is ...
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1answer
105 views

Integral of sum of sequence of integrable functions

Let $f_n$ be integrable function and $f_{n}^{+},f_{n}^{-}$ be its positive and negative parts. Are these steps correct? \begin{align} \int {\sum {f_n}} &=\int{\sum{(f_{n}^{+}-f_{n}^{-})}}\\ ...
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1answer
172 views

Every measurable homomorphism from $\mathbb{R}^n$ to $\mathbb{C}^*$ is exponential.

A standard result in real analysis says that if $f : (\mathbb{R}, +) \rightarrow (\mathbb{C}^*, *)$ is a Lebesgue measurable (group) homomorphism with $|f| = 1$, then $$(\exists ~\xi \in \mathbb{R})~ ...
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779 views

Folland, Chapter 1 Problem 17

Problem 17: If $\mu^*$ is an outer measure on $X$ and $\{A_i\}_{i=1}^{\infty}$ is a sequence of disjoint $\mu^*$-measurable sets, then $\mu^*(E\cap \cup_{j=1}^{\infty} A_j)=\sum_{j=1}^{\infty}(E\cap ...
3
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1answer
68 views

Lebesgue density for other probability measures on $[0,1]$

Does the Lebesgue density theorem hold for arbitrary (Borel) probability measures on $[0,1]$? Following Downey & Hirschfeldt's proof leads me to believe that the answer is "yes". (Recall every ...
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1answer
87 views

measurability w.r.t. Borel on extended real line

Following Schilling I have shown for measurable functions $$u, v \in m \mathcal{A}/ \mathcal{\hat{B}}$$ that sums, differences, products and maxima/minima are again measurable whenever they are ...
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1answer
254 views

Folland Proposition 1.13 Real Analysis, Second Edition

Proposition 1.13 (b) states: If $\mu_0$ is a premeasure on an algebra $\mathcal{A}$ and $\mu^*:\mathcal{P}(X)\to [0,\infty]$ by: $$\mu^*(E)=\text{inf}\left\{\sum_{j=1}^{\infty}\mu_0(A_j):\ ...
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161 views

Given $\mu$ the counting measure on an infinite set $\Omega$, $\lim \mu(A_n) \ne 0$

Problem: Let $\mu$ be the counting measure on an infinite set $\Omega$. Prove that there is a sequence of sets $A_1 \supset A_2 \supset A_3 \dots$ such that $\bigcap A_n = \varnothing$, but $\lim_{n ...
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1answer
176 views

About Borel bounded functions

I am studying the book of Reed and Simon, Functional Analysis and I am not able to prove the following exercise (16 on chapter 1): Prove that the bounded Borel functions on $[0,1]$ are the smallest ...
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1answer
73 views

question about Lebesgue's integral

let $f(x)$ be a bounded measurable function defined on $\mathbb{R}$, then define $$F(x)=\int_0^xf(t)dt,\ \ x\in\mathbb{R}$$ We can see that $F(x)$ is a absolutely continuous function and by some ...
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1answer
541 views

Conditions for Fubini's theorem

To preface this post, I have to admit that I have extremely little measure theory knowledge and I get lost when trying to read about Fubini's theorem for this reason. In the theorem statement for ...
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2answers
94 views

Holder's inequality and an infinite series question

I'm looking at a sample measure theory exam question. (a) State Holder's Inequality (b) Let $\{a_n\}$ be a sequence of non-negative real numbers and let $\epsilon\in(0,1)$ be such that ...
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0answers
92 views

Show that the following intervals are element of Borel sets

The set of numbers (the real line) is $\mathbb{R}$ = {x : -∞ < x < ∞ }. Events are Borel sets of $\mathbb{R}$. To have an idea which subsets of $\mathbb{R}$ are Borel sets, do the following. ...
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1answer
175 views

Compactness and compact-finite measure in Lusin theorem (Rudin)

I have two questions about some hypotheses in Lusin's theorem as stated in Rudin's "Real and Complex Analysis". The proof initially deals with a subcase, that is the function $f$ is supposed to be ...
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2answers
118 views

Is this a measurable space?

This probably has a very quick answer but it's been bugging me for a while. Take two copies of the Cartesian plane. On each plane cut out the square $S=(0,1)\times(0,1)$. Throw away the rest. Now ...
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0answers
49 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
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1answer
43 views

Is there a name for the following asymmetry property of a measure on $R$?:

Let $\mu$ be a Borel measure on $\mathbb{R}$. I am looking for a name for the following property: $\int_\mathbb{R} f d\mu \ge 0$ for all skew-symmetric Borel functions $f$ that are non-negative on ...
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141 views

About convergence of functions in $L_p$

Suppose we have a sequence of functions $f_n\ge 0$, $f_n\in L^p(\mu)$, $\int f_nd\mu=1$, and $f\ge 0$, $f\in L^p(\mu), \int fd\mu=1$, $0<p<1$ such that $$g_n:=\frac{f_n^p}{\int f_n^pd\mu}\to ...
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2answers
299 views

Constructing a complete measure space

Working on the following proposition in Royden: Proposition: If $(X,\mathcal{B},\mu)$ is a measure space, then we can find a complete measure space (one that contains all subsets of sets of measure ...
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1answer
474 views

$f$ measurable, $f=g$ almost everywhere, complete measure space [duplicate]

Let $X$ be a nonempty set, $\mathcal{X}$ a $\sigma$-algebra of subsets of $X$, and $\mu$ a measure on $\mathcal{X}$ (i.e., $\mu:X\to[0,+\infty]$, $\mu(\phi)=0$, and $\mu$ is countably additive. ...
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391 views

$f$ integrable but $f^2$ not integrable

At this point in Bartle, $X$ is a nonempty set, $\mathcal{X}$ is a $\sigma$-algebra of subsets of $X$, and $\mu$ is a measure on $\mathcal{X}$. $f\in L(X,\mathcal{X},\mu)$ means: $f:X\to R$ is ...
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193 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
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128 views

Naive example of integral with values in Banach space

Naive picture: If $X$ is a measurable space, the space $B$ of signed measures on $X$ is a Banach space. There is a natural map $f\colon X\to B$ (namely, $f(x)=\delta_x$). For any finite measure ...
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1answer
101 views

Why universally and not just Borel policies

In a famous book Stochastic Optimal Control: The Discrete-Time Case by Bertsekas and Shreve they use universally measurable policies that come up with some handy features: e.g. they show that every ...
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2answers
200 views

Small sets on $\mathbb{R}$.

I was thinking of different definitions of small subsets on $\mathbb{R}$, such as meagre or zero-measure. These are quite well-known, so I was searching for different notions. Define a set has ...
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1answer
146 views

Tricky detail in the proof of Haar's theorem

I'm trying to dig in the details of the proof of Haar's theorem, and at some point I need to use Fubini's theorem, which requires that if we want to change the order of integration over the product ...
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1answer
41 views

Show there exists an N such that $n\ge N$ implies $\int|f^+-\phi_n|\,d\mu<\epsilon/2$

Let $f\in L(X,\mathcal{X},\mu)$. This makes $\int f^+\,d\mu<+\infty$. Now, there exists a monotone increasing sequence of simple measurable functions $\phi_n$ that converge to $f^+$. By the ...
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108 views

Probability measure

Suppose $X\ge0$ a non negative random variable in (Ω,F,P) and $\int_Ω XdP=m<\infty$. Prove that $$ν(Α)=\dfrac{1}{m}\int_A XdP, with A\in F$$ is probability measure in (Ω,F,P). Do I have to ...