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

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3
votes
1answer
41 views

Haar measure of an angle-distance ball in SO(3)

If for rotations $R_0$, $R_1$ we define the distance $d(R_0, R_1)$ to be the angle of $R_0 R_1^{-1}$ and given $r\in [0,\pi]$, what is the "volume" (normalised Haar measure) in SO(3) of the ball ...
0
votes
3answers
16 views

Find a sequence of measurable functions defined on a measurable set $E$ that converges everywhere on $E$, but not almost uniformly on $E$.

Find a sequence of measurable functions defined on a measurable set $E$ such that the sequence converges everywhere on $E$, but the sequence does not converge almost uniformly on $E$. I'm having ...
3
votes
1answer
41 views

Measure of the set $\{(x+f(x), x-f(x)):x\in \mathbb R\}$ in $\mathbb R^2$ is 0

Let $f:\mathbf R\to \mathbf R$ be a Lebesgue measurable function. Then the set $S=\{(x+f(x), x-f(x)):x\in \mathbf R\}$ is Lebesgue measurable in $\mathbf R^2$ and its measure is $0$. I am ...
0
votes
1answer
17 views

$f_n \to f$ in $L^p$, $f_n\to g$ in $L^p{'}$ then $f=g$ a.e $x$.

If $f_n \in L^p\cap L^p{'}$ such that $p\neq p'$ and $f_n \to f$ in $L^p$, $f_n\to g$ in $L^p{'}$ then $f=g$ a.e $x$. a suggestion please.
1
vote
1answer
19 views

$\int_{\Omega\setminus A_n}f\;d\mu\to\int_\Omega f\;d\mu$ for all measurable $A_n\downarrow\emptyset$

Let $(\Omega,\mathcal{A},\mu)$ be a measure space $(A_n)_{n\in\mathbb{N}}\subseteq\mathcal{A}$ such that $A_n\downarrow\emptyset$, i.e. $A_n\supseteq A_{n+1}$ and ...
1
vote
1answer
34 views

This one wierd trick integrates fractals. But does it deliver the correct results?

It occurs to me that people most likely already know how to explicitly integrate over fractals, but my method seems to vastly simplify the process (So even a comparative layman like me can do it). ...
26
votes
2answers
8k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, ...
1
vote
1answer
24 views

The Essential Supremum as a Limit

Let $(X, \mathcal F, \mu)$ be a finite measure space and let $f\in L^\infty(X, \mu)$. Define $\alpha_n=\int_X |f|^n\ d\mu$. Then $$\lim_{n\to \infty}\frac{\alpha_{n+1}}{\alpha_n}=\|f\|_\infty$$ ...
1
vote
1answer
36 views

Does the integral converge I can't find counterexample

I found the following question in the book of kolomogorov fomin introductory real analysis and I don't know how to solve it. Does anyone have any ideas? Suppose $f$ is integrable on sets ...
1
vote
1answer
16 views

Measurability of function

Let $g: C[0,\infty) \to [0,\infty)$ be a Borel-measurable function. Define $f: C[0,\infty) \times C_0[0,\infty) \to C[0,\infty)$ by $f(y,z)(t)=y(t)+z(1-g(y))\mathbf{1}_{\{t > g(y)\}}$. In a proof ...
0
votes
0answers
22 views

proof that Riemann integrals is extended by Lebesgue integrals

After reading a proof sketch somewhere (forgot the link) I've written a proof in my own words. I'm not quite sure if I got the details right, since there were variants of this floating around that any ...
1
vote
2answers
167 views

Proving the Exterior (Outer) Measure of Rectangle is Equal to Volume

I'm having trouble understanding one step of Stein and Shakarchi's proof that the exterior measure of a rectangle is equal to its volume. The proof I reference is part of Example 4 in section 1.2 of ...
3
votes
0answers
21 views

A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
0
votes
1answer
10 views

Lower semicontinuous non-negative function on a locally compact Hausdroff space with a countable base

An extended real number is an element of $\mathbb R \cup \{-\infty, +\infty\}$. Let $X$ be a locally compact Hausdorff space with a countable base. An extended real valued function $f$ on $X$ is ...
2
votes
0answers
24 views

Joint Expectation of independent Random Variables given two sigma-algebras

We have a question regarding two random variables $X$,$Y$ on a probability space with sigma-algebra $\mathcal{F}$ and a sub-sigma algebra $\mathcal{M}$ such that $X$ is independent of $\mathcal{M}$ ...
1
vote
0answers
14 views

Nonzero Radon-Nikodym derivative invertible?

Suppose that $\nu$ is a $\sigma$-finite positive measure, and that $\rho$ is a measurable function that's nonzero $\nu$-a.e. Define $\mu(A) = \int_{A} \rho\, d\nu$ for all $\nu$-measurable $A$, so ...
1
vote
2answers
64 views

Do Riemann and Lebesgue integrals always agree?

I know that on a closed bounded interval, say $[a,b]$ in $R^1$, if a function is Riemann integral, then it is Lebesgue integrable, and the values of those two integrals are the same. But, is this ...
0
votes
1answer
16 views

Is it right to say that a positive measure is a signed measure by definition?

A signed measure $\mu$ is a measure which can also take on negative values. Now my question is, is a positive measure a special case of a signed measure since it essentially maps to a subset of ...
1
vote
1answer
23 views

Angle-doubling map is mixing

Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that $$ ...
0
votes
1answer
25 views

Basic finite dimensional distribution question

I'm having trouble wrapping my head around the basic idea of a finite dimensional distribution. Suppose $(\Omega, \Bbb P, \mathcal{F})$ is a probability space. Let $(X_{t})_{t \geq 0}$ be a ...
2
votes
2answers
108 views

Exact statement of the Radon-Nikodym Theorem

I am a bit confused about the exact statement of the Radon-Nikodym Theorem. Suppose that in the usual setup, $v \ll u$. Does it require both $v$ and $u$ to be sigma-finite, or only $u$ to be sigma ...
0
votes
0answers
17 views
0
votes
1answer
42 views

Why does a Borel measurable function imply its Lebesgue measure?

Borel measurable defined as: $f: D ->\mathbb R$ is Borel measurable if $D$ is a Borel set and if, for each real $a$, the set {$x∈D: f(x) > a$} is a Borel set. Definition of Lebesgue measurable ...
3
votes
1answer
27 views

Are measurable sets closed under projections?

For the following, let us assume that large enough sets to carry the arguments through do exist, i.e. that there are supercompact cardinals or whatever is sufficient. I know that all projective ...
0
votes
0answers
14 views

The mutual information rate spectrum

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...
0
votes
0answers
33 views

If $f_n \to f$ pointwise a.e., $\int |f| < \infty$, and if $\int |f_n| \to A$, is $A=\int |f|$?

We work on some domain $\Omega$ which may or may not be bounded. If $f_n \to f$ pointwise a.e., if $\int |f| < \infty$, and if we know that $\int |f_n| \to A$ to some number $A$, is $$A=\int ...
0
votes
0answers
25 views

Why do we need to declare a probability measure for the definition of stochastic processes?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be measurable with respect to ...
0
votes
0answers
27 views

Why the two expressions of total variation distance are equivalent?

In a stochastic processes textbook, I find the definition of total variation distance is $\|\pi - \nu\|_{TV} = \max\{|\pi(A) - \nu(A)|:A\subset S\}$ where $\pi$ and $\nu$ are two probability measures ...
4
votes
1answer
30 views

Lebesgue measure without choice

From this question and this question (and their answers) I gather that it is consistent with ZF without The Axiom of Choice to assume that there exist countable sets $A_n$, $n\in \mathbb N$, such that ...
1
vote
1answer
23 views

Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
0
votes
2answers
34 views

$f_1,f_2 \in L^q(\mu)$ and $\int_\mathcal{X}f_1gd\mu = \int_\mathcal{X}f_2gd\mu$ for all $g \in L^p(\mu)$ implies $f_1=f_2$ a.e.

Let $X=(\mathcal{X},\mathcal{M},\mu) $ be a measure space. Assume that $\mu$ is $\sigma$ finite and $1\leq p \leq \infty$, with $q$ the Holder conjugate exponent. If $f_1,f_2 \in L^q(\mu)$ and ...
0
votes
1answer
31 views

On the Preservation of Product Measurability under Partial Conditional Expectation.

Let $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ be probability spaces, $\mathcal{X}_{0}\subset\mathcal{X}$ a (sub)sigma field and assume that $f=f(x,y)\in L^{1}_{\mu\otimes \nu}$ where $(X\times ...
6
votes
1answer
350 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
1
vote
1answer
58 views

Is a countable intersection of open sets in $\mathbb R$ Lebesgue measurable?

If the answer is yes, how to prove that? Otherwise how to find a counterexample? Update: I've figured out the tricks inside. A countable intersection of open sets in $\mathbb R$ is equivalent to a ...
2
votes
0answers
39 views

How to learn problem solving strategy for Measure Theory?

I have taken both graduate level Algebra and Measure theory courses but found the latter much more difficult for me. I have put a lot effort on learning it by reading a few reference books and ...
2
votes
2answers
33 views

Showing that supremum function is integrable

Let $g_1(\omega),g_2(\omega),...$ be integrable functions defined on $\Omega$ with $g_n\rightarrow g$ and $g$ is integrable and also $\lim \int g_n=\int g$ . Define $h(\omega)= \sup_n g_n(\omega)$. ...
3
votes
0answers
39 views

An inequality for a maximal function on an $n$-ball.

We have $Mf(x) = \sup_{r>0} \frac{c_n}{r^n} \int_{|y|\le r} |f(x-y)| dy$ the maximal function, where $r^n/c_n$ is the volume of the n-dimensional ball of radius $r$, $|y|\le r$. I want to show ...
1
vote
0answers
13 views

Approximation of one function by other using a smooth multiplier function

This problem is from the Book, Harmonic Analysis by Katznelson (Problem 2, Page 160). Suppose $f$, $g\in L^2(\mathbb{R})$ such that $f(x) = 0$ implies $g(x)=0$ for almost all $x\in\mathbb{R}$. Then ...
0
votes
1answer
21 views

Outer Measures - Measure Theory

In the definition of an outer measure, they state the sub-additivity condition as $\mu_{*}(\bigcup A_{n}) \leq \sum\mu_{*}(A_{n})$ for any sequence of sets $A_{n} \subset X$ My question is does ...
0
votes
0answers
12 views

Definition of an Algebra - Measure Theory

So an algebra of a fixed set $X$ is a collection of subsets of $X$ such that it is closed under complementation and unions of sets. So is the difference between an algebra and a $\sigma$-algebra ...
2
votes
1answer
35 views

Application Birkhoff ergodic theorem

Let $(X,\mathcal{B},m,T)$ be a probability preserving transformation. Let \begin{align*} I:&=\{f\in L^1: f=f\circ T\}\\ B:&=\{g-g\circ T: g\in L^1\} \end{align*} I have to show that $$ ...
0
votes
0answers
16 views

Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
1
vote
1answer
36 views

Probability of tail event using Kolmogorov's 0-1 law

If $X_1,X_2,... $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I ...
2
votes
2answers
67 views

Lebesgue measure of graph of $\sin{\frac{1}{x}}$ on $[0,1]$

I am working on something and read that measure of graph of a continuous function on compact sets is zero. Now, I tried to do it for non continuous functions but the set of discontinuities have ...
1
vote
3answers
64 views

Compact subset of $\mathbb R$ whose Lebesgue measure is non-zero

Let $\mathbb R$ be the field of real numbers, $\mu$ the Lebesgue measure on it. Let $K$ be a compact subset of $\mathbb R$. Is the following assertion true? If $\mu(K) \gt 0$, then the interior ...
0
votes
0answers
8 views

Closed subgroup of a locally compact Hausdorff group whose Haar measure is non-zero.

Let $G$ be a locally compact Hausdorff group, $H$ its closed subgroup. To avoid pathologies, we assume the underlying topological space of $G$ has a countable base. Let $\mu$ be a Haar measure on $G$. ...
1
vote
0answers
15 views

Weil's definiton of image of Haar measure on homogeneous space $G/\Gamma$ where $\Gamma$ is discrete

Let $G$ be a locally compact Hausdorff group. To simplify matters we assume the underlying topological space of $G$ has a countable base. Let $\Gamma$ be a discrete subgroup of $G$, $G/\Gamma$ the ...
8
votes
2answers
114 views

Derivative of $\int_0^1 e^{\sqrt{x^2+t^2}}\,\mathrm{d}x$ at $t = 0$

Let the real-valued function $\phi:\mathbb{R}\to\mathbb{R}$ be defined by $$\phi(t)=\int_0^1e^{\sqrt{x^2+t^2}}\,\mathrm{d}x,$$ it can then be shown that $\phi$ is continuous and differentiable. I ...
1
vote
0answers
27 views

Defining Lebesgue measure on a subspace of $\mathbb{R}^n$

Let $\bar{w}_1,.., \bar{w}_k$ be linearly independent vectors in $\mathbb{R}^n$. Let $W$ be the subspace spanned by these $\bar{w}_i$'s. I know how the Lebesgue measure is defined on $\mathbb{R}^n$. ...
0
votes
1answer
41 views

What are the hypotheses in Levi's monotone convergence theorem?

Today I read monotone convergence theorem , dominated convergence theorem and fatou's lemma And I need some help We know the dominated convergence theorem in Measure theory In its proof we ...