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

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4
votes
1answer
57 views

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying: i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$. ii) there is some $\theta \in (0,1)$ ...
1
vote
0answers
42 views
+50

Following conditions for convergence of measures equivalent

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Let $\mu_n$ be a sequence of finite measures on $([0, 1], \mathcal{B})$ and let $\mu$ be another finite measure on $([0, 1], \mathcal{B})$. ...
2
votes
1answer
36 views

Does it follow that two finite positive measures are the same?

Suppose $\mu$ and $\nu$ are finite positive measures on the Borel $\sigma$-algebra on $[0, 1]$ such that $\int f\,d\mu = \int f\,d\nu$ whenever $f$ is real-valued and continuous on $[0, 1]$. Does it ...
1
vote
0answers
34 views

Show $N_p[f]=(\frac{1}{|E|}\int_{E}|f|^p)^{\frac{1}{p}}$ is monotone in $p$

For $0<p\leq \infty$ and $0<|E|<\infty$ ($|E|$ is the lebesgue measure of $E$), define $$ N_p[f]= \left( \frac{1}{|E|} \int_E |f|^p \right)^{1/p}, $$ where $N_\infty[f]$ means $\|f\|_\infty=...
3
votes
1answer
32 views

Does it follow that $\mu$ is a measure? [duplicate]

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Cal the limit $\mu(A)$. ...
0
votes
1answer
25 views

Prove that if $f(x)$ measurable on measurable set $E$ then $f^4(x)+7x^2$ measurable on $E$

Prove that if $f(x)$ measurable on measurable set $E$ then $f^4(x)+7x^2$ measurable on $E$ Attempt: for measurable functions $g(x),z(x)$the product $g(x)\cdot z(x)$ also measurable therfore $f^4(x)=...
3
votes
1answer
32 views

Alternate proof of the dominated convergence theorem by applying Fatou's lemma to $2g - |f_n - f|$?

Here is a proof of the dominated convergence theorem. Theorem. Suppose that $f_n$ are measurable real-valued functions and $f_n(x) \to f(x)$ for each $x$. Suppose there exists a nonnegative ...
7
votes
1answer
37 views

Variant of dominated convergence theorem, does it follow that $\int f_n \to \int f$?

Suppose $f_n$, $g_n$, $f$ and $g$ are integrable, $f_n \to f$ almost everywhere, $g_n \to g$ almost everywhere, $|f_n| \le g_n$ for each $n$, and $\int g_n \to \int g$. Does it follow that $\int f_n \...
4
votes
1answer
27 views

$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$ implies $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure and for some $\gamma > 0$, we have$$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty.$$Does it follow that $\{f_n\}$ is uniformly integrable?
7
votes
1answer
53 views

Does there exist a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges for each Borel subset $A$ of $[0, 1]$?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. Does there exist a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges for each Borel ...
3
votes
1answer
25 views

Does it follow that $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure, $f_n \to f$ almost everywhere, each $f_n$ is integrable, $f$ is integrable, and $\int |f_n - f| \to 0$. Does it follow that $\{f_n\}$ is uniformly integrable?
6
votes
1answer
71 views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
8
votes
2answers
49 views

$\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)...
4
votes
1answer
27 views

Countable collection of Borel subsets of $[0, 1]$, exists subsequence where $\int_A f_{n_j}(x)\,dx$ converges for each $i$?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $\{A_i\}$ is a countable collection of Borel subsets of $[0, 1]$, then there ...
1
vote
3answers
36 views

How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ ...
2
votes
1answer
21 views

seq. of nonneg. Lebesgue measurable functions on $\mathbb{R}$, have $\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx$?

Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $\mathbb{R}$. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$If not, ...
3
votes
0answers
32 views

What does “Borel space”, unqualified, refer to?

For examples of use, Google "in Borel space", without the quotes. I'm thinking it means either ℝ equipped with its Borel σ-algebra, or to Borel spaces in general (that is, topological spaces with a σ-...
0
votes
1answer
18 views

If two intervals are not disjoint, I can write them as a union of disjoint pieces

In this online lecture, the professor writes: $$E_1 \cup E_2 = (\sqcup_{i=1}^n I_j) \cup (\sqcup_{k=1}^n J_k) = \sqcup_{i=1}^n \sqcup_{k=1}^n(I_j \cap J_k)$$ where $I_j, J_k$ are intervals of $\...
1
vote
1answer
16 views

Is $\sigma(X,Y) = \sigma(X, X \cdot Y)$ for two Random Variables $X$ and $Y$?

Suppose we have two real random variables $X,Y$. Then clearly \begin{equation} \sigma(X, X \cdot Y) \subset \sigma(X,Y) \end{equation} since both $X$ and $X \cdot Y$ are $\sigma(X,Y)$-measurable ...
7
votes
1answer
69 views
+50

Intuition behind proof of bounded convergence theorem in Stein-Shakarchi

Theorem 1.4 (Bounded convergence theorem) Suppose that $\{f_n\}$ is a sequence of measurable functions that are all bounded by $M$, are supported on a set $E$ of finite measure, and $f_n(x) \to f(x)$ ...
4
votes
1answer
42 views

$f_n \to f$ almost everywhere and $\int |f_n| \to \int |f|$ implies $\int |f_n - f| \to 0$?

Suppose $f_n$ and $f$ are integrable, $f_n \to f$ almost everywhere, and $\int |f_n| \to \int |f|$. Does it necessarily follow that$$\int |f_n - f| \to 0?$$
2
votes
1answer
25 views

Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
2
votes
1answer
24 views

Sequence of nonnegative $f_n$ tending to $0$ pointwise where $\int f_n \to 0$, but there's no integrable function where $f_n \le g$ for all $n$?

What is an example of a sequence of nonnegative functions $f_n$ tending to $0$ pointwise such that $\int f_n \to 0$, but there is no integrable function such that $f_n \le g$ for all $n$?
2
votes
1answer
15 views

$f$ integrable, if either $A_n \uparrow A$ or $A_n \downarrow A$, then does it follow that $\int_{A_n} f \to \int_A f$? [closed]

Suppose $f$ is integrable. If either $A_n \uparrow A$ or $A_n \downarrow A$, then does it follow that $\int_{A_n} f \to \int_A f$?
0
votes
1answer
50 views

Infinite dimensional Borel-measurable function.

I am not quite sure how this statement about infinite dimensional borel-measurable functions is true, but the author says it is an easy observation: Let $D([0,\infty))$ denote the space of all ...
2
votes
0answers
30 views

What is an example of lower semicontinuous functions not satisfying this?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$. Let $u:X\rightarrow [0,\infty]$ be a lower semicontinuous function such that $\int_X u d\mu <\infty$. Then, does ...
2
votes
0answers
7 views

Can we approximate integrable functions by finite-valued upper,lower semicontinuous functions?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$ and $f\in L^1(\mu)$ be real-valued and $\epsilon>0$. Then, by Vitali-Caratheodory theorem, there exist upper ...
2
votes
0answers
27 views

Exchanging supremum and conditional expectation

I've come across a problem which seems similar to this but quite different and can't find a way of going around it. I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and ...
1
vote
3answers
68 views

Show that there is a subsequence of $(f_n)_n$ that converges to $f$ almost everywhere.

Let $(X,\mathcal{B}, \mu)$ be a measure space and assume the sequence $(f_n)_n$ converges to $f$ in $L^p(\mu)$, where $1\leq p<\infty$. Show that there is a subsequence of $(f_n)_n$ that converges ...
3
votes
2answers
120 views

sigma algebra generated by compacts versus sigma algebra generated by open sets

Let $\Omega$ be a locally compact Hausdorff set. Is the sigma algebra generated by compact sets is the same as the sigma algebra generated by open sets?
3
votes
2answers
47 views

L^1 convergence and limsup of convergent sequence

I have to solve this exercise: let $f_n$ be a sequence of positive real function defined on a measure space $(X,M,\mu)$ such that $f_n\in L^1(\mu)$ $\forall n\in \mathbb{N}$ and $f_n$ is convergent in ...
3
votes
2answers
23 views

If $f,g:X \to Y$ are measurable, is the set on which $f=g$ measurable? What $Y$ does this hold for?

If $f,g:(X,\Sigma_X) \to (Y,\Sigma_Y)$ are measurable, when can we conclude that $\{x \in X: f(x)=g(x)\} \in \Sigma_X$ is a measurable subset of $X$? This is a standard theorem when $Y=\mathbb{R}$ or ...
4
votes
0answers
60 views

Question on Young's inequality for convolution

Young's Inequality: Let $u\in {\cal{L}^1}(\lambda^n)$ and $v\in {\cal{L}^p}(\lambda^n)$ where $\lambda^n$ is the $n$ dimensional Lebesgue measure on $(\mathbb{R}^n,{\cal{B}}(\mathbb{R}^n))$ and $p\in[...
3
votes
1answer
58 views

Real Analysis, Problem 3.2.14 The Radon Nikodym Theorem

Problem 3.3.14 - If $\nu$ is an arbitrary signed measure and $\mu$ is a $\sigma$-finite measure on $(X,M)$ such that $\nu\ll \mu$, there exists an extended $\mu$-integrable function $f:X\rightarrow [-\...
4
votes
1answer
37 views

measure preserving map does not increase distance

I read a sentence saying "any measurable subset of $\mathbb{R}$ can be mapped to an interval by a measure-preserving transformation which does not increase distances" Here the measure is Lebesgue ...
3
votes
2answers
56 views

What is the infimum/supremum of a set in the extended real line?

May be a simple question, but I still don't know what the definition of the supremum or infimum of a set in the extended real line is. For example when we define the Lebesgue measure, we define the ...
0
votes
1answer
34 views

Continuity and convergence everywhere.

Suppose a sequence of continuous functions $(h_n)$ converges almost everywhere to another continuous function $h$ . Is it possible to infer that $h_n$ infact converges everywhere? If not, under what ...
1
vote
0answers
16 views

Question with Lebesgue-Stieltges outer measure

Let $g$ and $h$ be two increasing functions and $\theta_g$, $\theta_h$ be the associated Lebesgue-Stieltges outer measures on $R$ (the set of real numbers). We can also associate to $g+h$ the L-S ...
3
votes
1answer
44 views

Real Analysis, Folland Proposition 3.11 The Radon Nikodym Theorem

Proposition 3.11: If $\mu_1,\ldots,\mu_n$ are measures on $(X,M)$, there is a measure $\mu$ such that $\mu_j\ll\mu$ for all $j$ -- namely $\mu = \sum_1^n\mu_j$. Attempted proof: Suppose we have a ...
3
votes
2answers
42 views

Real Analysis, Folland Corollary 3.10 The Lebesgue Radon Nikodym Theorem

Background Information: Proposition 3.9 - Suppose that $\nu$ is a $\sigma$-finite measure and $\lambda$ are $\sigma$-finite measures on $(X,M)$ such that $\nu\ll \mu$ and $\mu\ll \lambda$. a.)...
1
vote
2answers
13 views

Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets

Okay so I have a measurable function $f$ and a set $E_{n,i}$ $$E_{n,i}=\left\{ x:\frac{i-1}{2^{n}} \leq f(x)<\frac{i}{2^{n}}\right\}$$ where $i=1,...,n2^{n}$ and $n=1,2,...$ Then I have another ...
-2
votes
0answers
32 views

Hölder inequality application to show that f=1 [closed]

I want to proof that if $f \in L^{1}_{\mu}(\mathbb{R}), f > 0$ continuous, satisfies $(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
4
votes
1answer
102 views

Conclusion about measurable functions from knowledge about continuous functions

Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb{R}$. We know that if $$\int f d\mu = \int f d\nu $$ for all continuous functions $f$ then $\mu=\nu$ and so the equation above holds for ...
1
vote
0answers
14 views

Measure induced by subgradient of convex functional

I am trying to understand why the following defines a measure. Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function. Define a measure $\mu$ on $\mathcal{B}(\mathbb{R}^d)$ by $$\mu(E) = \...
4
votes
1answer
65 views

Lebesgue integration by substitution

I read that, if $f\in L^1[c,d]$ is a Lebesgue summable function on $[a,b]$ and $g:[a,b]\to[c,d]$ is invertible and such that $g\in C^1[a,b]$ and $g^{-1}\in C^1[a,b]$, then $$\int_\limits{g([a,b])}f(x)\...
-1
votes
1answer
20 views

the relation between the sigma-algebras of two isomorphic spaces [closed]

It crosses my mind the following question : if X and Y are two isomorphic spaces what can we say about the Borel sigma-algebras associated to each of them, otherwise what is the relation between $\...
0
votes
0answers
15 views

$p$-power integral and $p$-series in higher dimensions

This seems like a basic question that should be addressed in a multivariable calculus course however I don't think I've ever confronted the issue until I became confused about a recent question here ...
6
votes
4answers
888 views

Is diameter of a set a measure?

Suppose the diameter of a nonempty set $A$ is defined as $$\sigma(A) := \sup_{x,y \in A} d(x,y)$$ where $d(x,y)$ is a metric. Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable ...
1
vote
1answer
28 views

Measure and probability : what would be $\mu(A\mid B)$?

I know that a probability space $(\Omega ,\mathcal F,\mathbb P)$ is in fact a measure space with a $\sigma -$algebra $\mathcal F$ and a measure $\mathbb P$. I know that if $A,B\in \mathcal F$ with $\...
0
votes
1answer
57 views

Real Analysis, Folland Corollary 3.6, The Lebesgue-Radon-Nikodym Theorem

Background Information: This is a Corollary to Theorem 3.5, found here. If $\mu$ is a measure and $f$ is an extended $\mu$-integrable function, the signed measure $\nu$ defined by $\nu(E) = \int_{E}...