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

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A question about “nice” functions

Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ...
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96 views

Exercise on Lebesgue measure ( Treatise of Analysis Vol2 by Dieudonné)

Someone challenge me to bring the solution from anywhere! So I have posted here and see, I am optimist because this website is excellent and its members are so helpful. Let me start with this ...
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46 views

$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$

For $f\in L^{p}$, $p \in [1,\infty)$ we want to prove: $$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$ I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
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103 views

Isomorphism Subalgebra

Given, the unit interval $I$ endowed with the Lebesgue measure $\mu$, and let $A$ be the (Boolean) algebra of Jordan measurable subsets of $X$ with respect to $\mu$, (i.e. those sets that satisfying ...
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100 views

Counterexample to upper continuity

Let $M$ be a $\sigma$-algebra of subsets of a set $X$ and let $\mu:M\rightarrow[0,\infty)$ be a finitely additive set function. I'm trying to decide if it's automatically true that for all ascending ...
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20 views

$f\in L^1(K)$ implies $f\in L^\infty (K_0)$ and $\int_{K\setminus K_0}f\leq\delta$, $\delta>0$ and some $K_0$.

Suppose that $K\subset\mathbb{R}^N$ is a measurable set and $f\in L^1(K)$ with $f\geq 0$. For given $\delta>0$, is it possible to find a measurable set $K_0\subset K$ such that $f\in L^\infty ...
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53 views

Sufficient conditions for $z \to \int_\mathbb{R} h(z,x) \,d\mu(x)$ to be analytic

Let $\mu$ be a measure on $\mathbb{R}$ and $G$ an open subset of $\mathbb{C}$. Every function $h \,:\, G\times\mathbb{R} \to \mathbb{C}$ then gives rise to a function $$ F_h \,:\, G \to \mathbb{C} ...
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478 views

Measure and the intersection with an open interval

Let $E \subset \Re$ be measurable with $\mu(E) > 0$. Show that for every $0 < \epsilon < 1$ there is an open interval $I$ such that $\mu(E \bigcap I) > (1-\epsilon)\mu(I)$ Let $0 ...
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89 views

Vitali covering problem

Let $ R $ denote the set of rational numbers in $ [0,1] $ and for each $ r∈R $, let $ V_r = \{ [r,r+ 1/k] : k=1,2,3,\ldots\} $. Put $V = ⋃_r V_r $. Show that for every $ ε > 0 $ there exists a ...
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43 views

why S' is closed with respect to the formation of finite unions?

$Proposition$ : Let S be a semiring of subsets of a set X. Define S' to be the collection of unions of finite disjoint collections of sets in S. Then S' is closed with respect to the formation of ...
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158 views

$f$ integrable $\implies g(x) = \int_{-\infty}^x f$ is absolutely continuous

Suppose that $f : \mathbb{R} \to \overline{\mathbb{R}}$ is an integrable function. Show that the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x) = \int_{-\infty}^x f$ is absolutely ...
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59 views

Question in Lebesgue integrable functions.

Suppose $g$ be a measurable function satisfying: $∀$ $σ∈[c,d]$ , there exists $δ>0$ such that $∫_E|g| <∞$ where $E=[σ-δ, σ+δ]$. Prove that $g$ is Lebesgue integrable on $[c, d]$.
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176 views

Approximating measurable functions on $[0,1]$ by smooth functions.

Let $f$ be a measurable function on $[0,1]$. Is there a sequence infinitely differentiable $f_n$ such that one of $f_n\rightarrow f$ pointwise $f_n\rightarrow f$ uniformly ...
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65 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
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51 views

Measurable Set From Cauchy sequences

Suppose that $D$ is a measurable set and that for each integer $n \geq 1$, $f_n : D \to \mathbb{R}$ is a measurable function. Prove the set $$E = \{x \in D \mid (f_n(x))_{n \geq 1} \text{ ...
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55 views

Properties of Lebesgue functions

If $f\in \mathcal {L}$ then there exists a sequence $\{f_k\}$ of step functions s.t. $\lim_{k\to\infty} f_k(x)=f(x)$ for almost all $x$ and $$\lim_{k\to\infty} \int|f(x)-f_k(x)|\,dx=0.$$ If I have ...
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198 views

Outer measure defined with rectangles

I'm studying Measure Theory by myself and I would appreciate some guidance about my proof. My textbook constructs an outer measure as following: $$m_*(E)=\inf\sum_{k=1}^{\infty}|Q_k|$$ where the ...
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113 views

Pairwise measurable derivatives imply measurability of combined derivative

I've found the following simple claim in an article. Unfortunately, i don't understand the proof given there nor can i come up with an alternative proof of my own. Maybe math.stackexchange can give me ...
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238 views

Every Lebesgue measurable function with bounded support is nearly bounded.

Let $f$ be a Lebesgue measurable function over the (non extended) reals with bounded support. I was wondering if we can say that, for every $\epsilon > 0$ there exists a bounded function $g$ such ...
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115 views

Show that $B(R^d)$ is the smallest $\sigma$-algebra satisfy the condition

Show that collection of all Borel set in $R^d$ (i.e. $B(R^d)$) is the smallest $\sigma$-algebra which make all continuous functions on $R^d$ measurable
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64 views

Prove there is a Borel measure u such that $u[x,y) = a(y) - a(x)$

If anyone has a solution to the following exercise, I would be grateful. Thanks. Let $\alpha$ be continuous and increasing on $[a,b]$. Prove that there exists a unique Borel measure $\mu$ on ...
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26 views

If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?

This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it. Consider a probability measure space $(X,\Sigma,\mu)$ and ...
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44 views

Is $\frac{|C\cap B_r(x)|}{|B_r(x)|}$ decreasing in $r$?

Suppose $C$ is a measurable set, $x\in C$, is $$ \frac{|C\cap B_r(x)|}{|B_r(x)|} $$ decreasing in $r$? Or any counterexamples? Thanks! Edit: @user39992 and @Karolis Juodelė show that it can not ...
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21 views

How to show $\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} F$ for any set $F \subset \mathbb{R}$ and $f$ continuously differentiable?

Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable with continuous derivative. I have to show that for all sets $F \subset \mathbb{R}$, the inequality $$\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} ...
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197 views

Uniformly Integrable of sets in $L_{1}(\mu)$ is equivalent to almost order boundedness

A bounded set $F\subseteq L_{1}(\mu)$ is said to be uniformly integrable if : $\forall \epsilon$ there is a $\delta>0 $, such that $\forall$ measurable set $A$, and $\forall f\in F$ , if ...
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144 views

Counter example to an adaptation of the Riesz-Markov theorem.

Suppose that $(K,\tau)$ is a topological space and that $\phi$ is a positive linear functional on $C(K)$. Then is it true that there exists a unique Baire measure $\mu$ on $K$ such that $\phi(f) = ...
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92 views

Uncountable sum vs Riemann Integral

Say we have $f(x)=1$ for $x\in \Re $ $\sum_{x\in [0,1]} f(x)=\infty$ but $\int_{[0,1]}f(x)dx=1$ I thought it was intuitive to think that integrals are just representations of infinite sums but I'm ...
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122 views

Suppose $f \colon \Bbb R \rightarrow \Bbb R $ and that $f$ is increasing. Show that $f$ is measurable?

Suppose $f \colon \Bbb R \rightarrow \Bbb R $ and that $f$ is increasing. ($x$ < $x'$ $\implies $ $f(x)$ < $f(x')$). Show that $f$ is measurable. I am a self taught person and just started ...
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207 views

Fail of optional sampling theorem

Could anyone help me see why the optional sampling theorem ($E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$ a.s.) fails for certain stopping times $\sigma\leq\tau$ for the not uniformly integrable ...
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334 views

Is the product of two measurable subsets of $R^d$ measurable in $R^{2d}$?

Suppose that $E_1,E_2$ are two measurable (Lebesgue) subsets of $R^d$. Define $E=E_1\times E_2=\left\{(x,y)|x\in E_1, y\in E_2\right\}$. Can we say that $E$ is a Lebesgue measurable subset of ...
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993 views

Doob's supermartingale inequality

I'm trying to prove that For a non-negative supermartingale $M$ it holds that for all $\lambda>0$ we have $$\lambda P\{\sup_{n}M_{n}\geq\lambda\}\leq E(M_{0})$$ My idea was to use Markov's ...
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325 views

If $f_n$ converges uniformly to $f$ on a measure space, show integral of $f_n$ converges to integral of $f$.

Please help me with this problem! Let $(\Omega,\cal F, \mu)$ be a measure space on which $(f_n)$ is a sequence of bounded, measurable, real-valued functions converging uniformly to $f$. If ...
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156 views

Lebesgue integrable function and limit

Show that if $f$ is a Lebesgue integrable function on $A\subset\mathbb R$ and $$A_n=\{x\in A:|f(x)|\geq n\}$$ for $n\in\mathbb N$, then $\lim_{n\to\infty} n\cdot m(A_n)=0$. My solution which is ...
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249 views

Why does the supremum over finite partitions not suffice in defining total variation of complex measure?

In Rudin's Real and Complex Analysis, Chapter 6, eqn. 3, the total variation of a complex measure is defined as a supremum over all possible partitions of a set. Why do we need to consider all ...
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170 views

Egoroff's Theorem for extended real-valued functions

Suppose $\{f_n(x)\}$ is a sequence of extended real-valued functions. I say that this sequence converges uniformly to $f(x)$ provided for every $\epsilon > 0$ there exists $N$ such that if $n > ...
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109 views

A question about $L_p$ spaces and inclusion.

There is a corollary in my textbook that says, Let E be measurable, $m(E) < \infty$ , and $1 \leq p_1 < p_2 \leq 1$. Then $L_{p_2}(E) \subseteq L_{p_1}(E)$. Furthermore, $||f||_{p_1} ...
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90 views

Measure nonzero implies dense on a rectangle

This would be a very handy lemma for me but I have been unable to prove it thus far. If $S \in \mathbb{R}^n$ is bounded and is not of measure zero, then there exists a rectangle $R$ such that $S$ ...
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63 views

what does $(\Omega^T,\mathcal{A}^T)$ mean?

Let $(\Omega_t,\mathcal{A}_t), t\in T$ be a collection of measurable spaces. What does the notation mean? $(\Omega^T,\mathcal{A}^T)$
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88 views

How to understand $\frac{dP}{dQ}$, where $P, Q$ denote two distributions?

I am currently reading a paper named Estimating Individualized Treatment Rules Using Outcome Weighted Learning by Zhao et al., where they wrote an equation $$\frac{dP^D}{dP}=\frac{I(a=D(x))}{P(A=a)}$$ ...
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67 views

Lorentz Space property: $\left\| f \right\|_{L^{q,s}} \leq \lim\limits_{n\to\infty} \| f_n \|_{L^{q,s}}$

I would like to understand a statement similar to Fatou's Lemma in the Lorentz space setting. It is as follows. Suppose $0 < q,s < \infty$ and $f_n,f$ are measurable functions on a ...
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111 views

Measurable if and only if absolutely convergent

Let $N$ be the set of natural numbers, $M = 2^N$, and $c$ the counting measure defined by setting $c(E)$ equal to the number of points in $E$ if $E$ is finite and $\infty$ if $E$ is an infinite set. ...
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33 views

A question on a property of a nonnegative regular Borel measure.

I am reading through a proof and ran across this statement, and I would just like clarification. Suppose $\mu$ is a nonnegative regular Borel measure on $(\mathbf{R}^n)^m = \mathbf{R}^n \times ...
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146 views

$L_p$ spaces and convergence

The Riesz-Fischer Theorem implies that Lp-convergence implies pointwise a.e. convergence of a subsequence. There is an example that shows that the converse may not be true... Let E = [0, 1], $1 ...
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96 views

How to show measurability of a function implies existence of bounding simple functions

If $(X,\mathscr{M},\mu)$ is a measure space with $\mu(X) < \infty$, and $(X,\overline{\mathscr{M}},\overline{\mu})$ is its completion and $f\colon X \to \mathbb{R}$ is bounded. Then $f$ is ...
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47 views

Question about $\sigma$-Algebra

I have this definition : after that in order to prove a theorem (with $\mu \geq 0$ $\sigma$-finite $\mathcal{T}$ complete ). they say "Remark that $\mu$ is equivalent to a bounded measure so ...
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368 views

Measurable with respect to counting measure

$\Bbb N$ is the set of natural numbers. Let $(X, A, \mu) = (Y, B, \nu) = (\Bbb N, M, c)$ be the measure space such that $M = 2^N$, and $c$ the counting measure defined by setting $c(E)$ equal ...
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220 views

Lebesgue Integral on Lebesgue measurable set satisfies Caratheodory condition

Let $f$ be a non-negative, measurable, and integrable over every compact set in $\Omega$, where $\Omega$ is an open set $\subset \mathbb{R}^d$. For every Lebesgue measurable set $E$ (abbreviated as ...
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54 views

Independence of $n$ random variables

Let $A_1,A_2,\ldots,A_n$ be independent subsets of probability space $(\Omega, \Sigma, P)$ (For every $I\subseteq \{1,2,\ldots,n\}$, $P(\bigcap_{j\in J}A_j)=\prod_{j\in J}P(A_j) )$. Prove that ...
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199 views

Is the Lebesgue integral of a continuous function necessarily continuous?

Suppose that $f:\mathbb R \rightarrow [0,\infty]$ is measurable and that $\int_{\mathbb R}f \, dx<\infty$. (a) Prove that $F(x)=\int^x_{-\infty}f(y) \, dy$ is a continuous function. (b) Is $F$ ...
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98 views

cdf induced by a cdf

Let $(E,\Sigma, \mathbb{P})$ be a probability space, and $X: E\to \mathbb{R}$ a random variable. $F$ is the cdf of $X$. Define a new random variable $Y:=F(X)$. What is the cdf of $Y$?