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

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7
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0answers
45 views

Maximal ideals in the ring of measurable functions

The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,...
1
vote
0answers
115 views

Switching independent experiments: Does $I_A(X_1,\dots,X_n,Y_{n+1},Y_{n+2},\dots)$ converge almost surely?

Suppose there is a sequence of independent random variables $X=(X_1,X_2,\dots)$ with $X_i$ taking values in some arbitrary measure space $E_i$. Now there is a second sequence $Y=(Y_1,Y_2,\dots)$ ...
0
votes
1answer
29 views

For an invertible measure preserving system, $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$

For an invertible measure preserving system, show that $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$. Here we consider the measure preserving system $(X,\mathcal A,\mu,T)$ where $T$ is invertible and $\mu$...
0
votes
0answers
34 views

Some counterexamples on measurable function about continuity and discontinuity.

By Lusin, in finite measure, measurable function is almost continuous, i.e., except small measure, the function is continuous. What if we remove the condition 'finite measure'? Is there a measurable ...
1
vote
1answer
41 views

Real Analysis, Folland Proposition 2.13 Integration of Nonnegative Functions

Question: Proposition 2.13 - Let $\phi$ and $\psi$ be simple functions in $L^+$. a.) If $c\geq 0$, $\int c\phi = c\int \phi$. b.) $\int(\phi + \psi) = \int \phi + \int \psi$. c.) If ...
1
vote
1answer
39 views

Real Analysis, Folland Problem 2.1.5 Measurable Functions

Problem 2.1.5 - If $X = A\cup B$ where $A,B\in M$, a function $f$ on $X$ is measurable if and only if $f$ is measurable on $A$ and on $B$. Proof - Suppose, $X = A\cup B$ where $A,B\in M$ and we have ...
3
votes
5answers
223 views

properties of distributions

If $$\int_{-\infty}^\infty f dx = 1$$, with $f > 0 \forall x$, then prove or disprove: $$\int_{-\infty}^\infty \frac{1}{1 + f} dx $$ diverges. The hint I got is to consider the measure of the set$(...
1
vote
2answers
51 views

Real Analysis, Folland Problem 2.1.2 Measurable Functions

Exercise 2 - Suppose $f,g:X\rightarrow \overline{\mathbb{R}}$ are measurable. a.) $fg$ is measurable (where $0\times(\pm\infty) = 0$). b.) Fix $\alpha\in\overline{\mathbb{R}}$ and define $h(...
0
votes
1answer
21 views

sigma algebra generated by fraction of random variables (continued) [closed]

Suppose $X_1,X_2,X_3$ are positive i.i.d. random variables. Let $S=X_1+X_2+X_3$. Is this true that $$ \sigma(X_1,X_2)\subset\sigma(X_1/S,X_2/S)? $$ Any hint of this will be appreciated. Thanks.
2
votes
2answers
77 views

$L^p \subset L^q$

Let $(X,M,\mu)$ be a measure space. Let $\Omega \subset X$ be a measurable set. We have $L^2(\Omega) \subset L^1(\Omega)$ . Can we have that $\mu(\Omega)< \infty $ ?
3
votes
0answers
37 views

Comparing different definitions of tightness for measures

Let $X$ be a Hausdorff space, $\mathcal{B}(X)$ the Borel $\sigma$-algebra and $\mu : \mathcal{B}(X) \to [0, \infty]$ a measure. Consider the following properties: (1) $\forall A \in \mathcal{B}(X): \...
0
votes
1answer
46 views

Convergence in $L^p$ and convergence almost everywhere

Why $f_n$ converges to $f$ in $L^p$ space implies that exists subsequence of $f_n$ converging to $f$ almost everywhere?
1
vote
1answer
34 views

relation between measurable function and continuous functions

what is the relation between measurable function and continuous function? Which one implies the other? Examples for both type of functions. Whether sinx, cosx, tanx etx are measurable functions on ...
2
votes
1answer
38 views

Real Analysis, Folland Proposition 2.11/Exercise 10 Measurable Functions

Question Proposition 2.11 (Exercise 10) - The following implications are valid if and only if the measure is complete: a.) If $f$ is measurable and $f = g$ $\mu$-a.e., then $g$ is measurable. ...
0
votes
2answers
33 views

Addition of two measurable sets

Notation: $ A+B = \{ a + b : a \in A, b \in B \}. $ H. Steinhaus proved the classical result that $ A+B $ contains an interval if $ A $ and $ B $ are both measurable subsets of the real line, each ...
0
votes
1answer
39 views

Intermediate Value Like Property for Lebesgue Measure

Below is a question from N.L. Carother's book Real Analysis. I've provided my attempt at a solutions, however, any feed back would be very appreciated. Suppose $E$ is a measurable subset of $\...
2
votes
1answer
39 views

Can anyone explain the connection between reverse fatou's lemma and Fatou's lemma?

Here is the version of reverse Fatou's lemma I am looking at. $E_n$ is a sequence of events. $P(\limsup E_n) \geq \limsup P(E_n)$ Here is Fatou's lemma. Let $f_1,f_2,\ldots$ be a sequence ...
0
votes
1answer
19 views

Definitions of inner and outer regular measures

The definitions can be found here. I am trying to figure out why it's defined specifically this way. More precisely- Why compact sets are used to define inner regularity while open sets are used for ...
2
votes
1answer
42 views

Does the image of positive measure set under homeomorphism also have positive measure?

Say I have a homeomorphism $f:A\longrightarrow B$ between open subsets $A$ and $B$ of $\mathbb{R}^n$. If $S\subset A$ has positive Lebesgue measure, does $f(S)$ also have positive measure? If so, do ...
2
votes
1answer
50 views

Why the space of complex measures is Banach?

I've read the proof from here: Space of Complex Measures is Banach (proof?) and understood the part that proves that constructed limit is complex measure. But the first part is a bit unclear for me. I ...
-1
votes
1answer
32 views

limit superior and law of large numbers [closed]

I am wondering whether the following result is true: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. real-valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ ...
1
vote
0answers
18 views

Characterization of Banach sublattice of L^1

Let $(X, \Sigma, \mu)$ be a measure space and let $F\subset L^1(X,\Sigma,\mu)$ be a Banach sublattice of $L^1$ with the following properties: (1) If $f\in F$, $f$ real-valued, then $f\land 1\in F$ (...
2
votes
1answer
44 views

Characterization of measures such that $\frac{1}{x} \in L^1(H)$

Let $H$ be a finite measure on $(0,1)$. What conditions must $H$ fulfill, such that \begin{equation*} \frac{1}{x} \in L^1(H),\ \ \ \frac{1}{1 - x} \in L^1(H) \end{equation*} I'm trying to characterize ...
1
vote
0answers
19 views

Alternative definition of Lebesgue measurable set.

Consider $\mathbb{R^d}$ with Lebesgue measure $\mu$. Suppose that for any $\epsilon >0$, there exists a Lebesgue measurable set $F$ such $\mu^* (E\Delta F) < \epsilon.$ Then $E$ is Lebesgue ...
0
votes
1answer
60 views

Radon Nikodym Thm: extending to $\sigma$-finite case

I am reading Bartle's "Elements of Integration". Radon-Nikodym Thm: Let $\lambda,\mu$ be $\sigma$-finite measures on a measurable space $(X,\textbf{X})$ and say $\lambda \ll \mu$. Then $\exists$ ...
1
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0answers
22 views

Sets not in a sigma-algebra

I have a question concerning some sets which are not in a given sigma-algebra. More precisely, I have two questions closely related: Let $\mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d), d \ge 1$, be ...
1
vote
0answers
30 views

Well-definedness of Fourier transform of $f\in L^1(\mathbb{R})$

Let $f\in L^1(\mathbb{R})$ where the measure is taken to be the Lebesgue measure. The Fourier transform of $f$ is the function $\hat{f}$ defined as $$\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{...
2
votes
3answers
80 views

Real Analysis Folland, Proposition 2.6 Measurable functions

Question: Proposition 2.6 - If $f,g: X\rightarrow \mathbb{C}$ are $M$-measurable, then so are $f + g$ and $fg$. Attempted proof/brainstorm - Suppose $f,g: X\rightarrow \mathbb{C}$ are $M$-...
2
votes
0answers
26 views

Show linearity of a functional if it holds for nonnegatives

Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space $(X,\...
2
votes
2answers
39 views

Finite Borel measure on a compact Metric Space

Suppose that $\mu$ is a finite Borel measure on a compact metric space $X$ and that $\mu(\{x\}) = 0$ for all $x\in X$. Show that for every $\epsilon > 0$ there is a $\delta > 0$ such that for ...
0
votes
0answers
16 views

A question regarding the proof of Riesz Representation Theorem for the dual of Lp.

The theorem is stated below. Pasted following is the proof. Can anyone explain why $g \rightarrow S(g)-\int_X fgd\mu \ for\ all\ g\in L^p$ is continuous? And right above equation 15, why is $|g_n -...
0
votes
0answers
14 views

Can anyone explain one step in the proof of Helley's Theorem?

The theorem and its proof is pasted at the end of the post. Can anyone why this limit holds? What is a good way to think of it intuitively? $$lim_{k \rightarrow \infty} T_{n_{k}}(f_j) \ for\ all\ j?$$...
2
votes
1answer
31 views

Random set of rationals topological properties

Flip a coin (probability of heads is p, strictly greater than 0 and strictly less than 1) for every rational number. For each toss, if heads include the number in a set S, if tails exclude it. What is ...
2
votes
1answer
41 views

Capacity of a set in $\mathbb{R}^n$

The $2$-capacity of a set $\Omega$ sitting inside an open set $V \subset \mathbb{R}^n$ is given by $$\text{cap}_2(\Omega, V) = \inf_{u \in C^\infty_0(V), u|_\Omega \equiv 1} \int_V |\nabla u|^2 dx.$$ ...
1
vote
1answer
98 views

Infimum of lower semicontinuous functions

The following proposition is from the book Nicolae Dinculeanu Integration on Locally Compact Spaces: Let $H$ and $K$ be two compact Hausdorff spaces and $\alpha$ a continuous mapping of $H$ onto $K$. ...
0
votes
0answers
66 views

What does it mean for a function to be invertible 'almost everywhere'

It seems to me that the correct definition of a measure-theoretic inverse for a function f is a function g such that $f \circ g$ and $f \circ g$ are the identity almost everywhere. The problem I have ...
3
votes
0answers
61 views

Integral over Julia Set (Is my math correct?)

So I was answering this question about whether or not the Julia Set was self-similar in a known way. Of course it is, and that got me thinking. Even though the self similarity is nonlinear, what if ...
1
vote
1answer
28 views

Can anyone explain one step in the proof of Riesz Representation theorem?

I am trying to understand the following lemma from Royden's Real Analysis which is directly used to prove Riesz Representation theorem. The book in the proof states that " when $p = 1$. We must ...
1
vote
3answers
66 views

Real Analysis, Folland Problem 1.5.30 Borel measures

If $E\in L$ and $m(E) > 0$, for any $\alpha < 1$ there is an open interval $I$ such that $m(E\cap I) > \alpha m(I)$. Attempted proof/brainstorm - Let $E\in L$ with $m(E) > 0$ and suppose ...
3
votes
2answers
48 views

If function is measurable on an interval, is it measurable on its subinterval?

This is exercise 2.3 from "A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson: Let $[c,d]\subseteq[a,b]$. Show that if $f$ is measurable on $[a,b]$, then $f$ is ...
0
votes
1answer
29 views

Sequence of subsets $E_n$ of $[0, 1]$ with $m(E_n) = 1$ for every n but $m(\cap_{n=1}^\infty E_n) \neq 1$

This is actually a "prove or give counterexample" type of problem. The claim is that if any sequence of subsets $E_n$ of $[0, 1]$ have $m(E_n) = 1$ (m being the standard Lebesgue measure) for every n, ...
1
vote
1answer
49 views

Real Analysis, Folland Problem 1.5.29 Lebesgue measurable set

1.5.29 - Let $E$ be a Lebesgue measurable set. a.) If $E\subset N$ where $N$ is the nonmeasurable set described in section 1.1, then $m(E) = 0$. b.) If $m(E) > 0$, then $E$ contains a ...
2
votes
1answer
29 views

Real Analysis Folland Problem 1.5.28 Borel Measures

Problem 1.5.28 - Let $F$ be increasing and right continuous, and let $\mu_F$ be the associated measure. Then $\mu_F(\{a\}) = F(a) - F(a-)$, $\mu_F([a,b)) = F(b-) - F(a-)$,$\mu_F([a,b]) = F(b) - F(a-)$,...
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votes
0answers
34 views

Center of a mass for linear $f$ over $E$ with $(x,y,z)\in E \iff (x\cos\theta-y\sin\theta,x\sin \theta+y\cos\theta,z)\in E$

Let $E\subset \Bbb{R}^3$ be a measurable set (i.e. $\int_{\Bbb{R}^n}1_{E}$ exists) and let $v(E)\ne 0$. Let $f$ be a linear function $f:\Bbb{R}^3\to \Bbb{R}$, and let $(x_0,y_0,z_0)$ be the center of ...
2
votes
0answers
39 views

Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
1
vote
2answers
28 views

The Lebesgue measure of the set of horizontal lines through the points of a subset $A$ of $\mathbb{R}$ with $\lambda(A)=0$

Suppose $A$ is a subset of the real line with $\lambda(A)=0$ and $H=\{(x,y):x\in A\}$. What is a natural idea behind proving that $\lambda(H)=0$ ? In fact, I wish to prove that the collection of ...
2
votes
2answers
36 views

Bound on integrable nonnegative function $F$ given inequality with compactly supported continuous functions.

Full Question: Suppose that $F$ is a nonnegative function that is integrable on $\mathbb R$ and there is a constant $C$ such that $\int_\mathbb R Ff \leq C\int_\mathbb R f$ whenever $f$ is a ...
0
votes
0answers
34 views

Sufficient (and necessary?) conditions for normality of Gaussian process integral

Question Let $X(\cdot)$ be a Gaussian process on $\mathcal{J}=[a,b]\subseteq\bar{\mathbb{R}}$ (extended real line) with mean $\theta(\cdot)$ and covariance $\Sigma(\cdot,\cdot)$. My goal is to find ...
0
votes
0answers
23 views

What is a sequence of functions that converge weakly in Lp, but not strongly?

I am reading Royden’s real analysis. In his book, a sequence of functions in Lp converges weakly if every bounded linear functional in the dual space converges in R. Can anyone discuss the ...
1
vote
0answers
23 views

Show that $f$ defined on the interval $(a,b)$ is not differentiable for every point in $E$ with $m(E)=0$

Let $E$ have measure zero contained in the open interval $(a,b)$. In a previous problem I showed that there is a countable collection of open intervals, $\{(c_k,d_k)\}_k$, contained in $(a,b)$ for ...