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

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2
votes
0answers
37 views

If $F$ is a closed subset of $[0, 1]$, then how do I see that there exists a finite measure on $[0, 1]$ whose support is $F$? [closed]

If $X$ is a metric space, $\mathcal{B}$ is the Borel $\sigma$-algebra, and $\mu$ is a measure on $(X, \mathcal{B})$, then the support of $\mu$ is the smallest closed set $F$ such that $\mu(F^\text{c}) ...
5
votes
1answer
47 views

What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?
4
votes
1answer
31 views

Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$? [closed]

Suppose $m$ is Lebesgue measure. Define $x + A = \{x + y : y \in A\}$ and $cA = \{cy : y \in A\}$ for $x \in \mathbb{R}$ and $c$ a real number. Let $A$ be a Lebesgue measurable set. I have two ...
3
votes
1answer
18 views

Example of Lebesgue measurable subsets satisfying conditions [closed]

Let $m$ be Lebesgue measure. What is an example of Lebesgue measurable subsets $A_1, A_2, \ldots$ of $[0, 1]$ such that $m(A_n) > 0$ for each $n$, $m(A_n \Delta A_m) > 0$ if $n \neq m$, and $m(...
3
votes
1answer
27 views

Lebesgue-Stieltjes measure corresponding to a right continuous increasing function, $m(\{x\}) = \alpha(x) - \alpha(x-)$ for each $x$

Let $m$ be Lebesgue-Stieltjes measure corresponding to a right continuous increasing function $\alpha$. How do I see that for each $x$, we have$$m(\{x\}) = \alpha(x) - \alpha(x-)?$$
2
votes
2answers
79 views

What kind of proof is this? - Examining all the possibilities.

To prove the following : If $f : X \rightarrow Y$ is a measurable function, and $E$ is a Borel set, then $f^{-1} (E)$ is a measurable set. Prove) First, define $\Omega$ a collection of all $E \subset ...
3
votes
0answers
34 views

Showing that $\sigma$-algebra is uncountable [duplicate]

Suppose $\mathcal{A}$ is a $\sigma$-algebra with the property that whenever $A \in \mathcal{A}$ is nonempty, there exist $B$, $C \in \mathcal{A}$ with $B \cap C = \emptyset$, $B \cup C = A$, and ...
3
votes
0answers
25 views

Is $\bigcup_{i = 1}^\infty \mathcal{A}_i$ necessarily a $\sigma$-algebra? [duplicate]

Suppose $\mathcal{A}_1 \subset \mathcal{A}_2 \subset \ldots$ are $\sigma$-algebras consisting of subsets of a set $X$. Is $\bigcup_{i = 1}^\infty \mathcal{A}_i$ necessarily a $\sigma$-algebra? If not, ...
6
votes
1answer
23 views

Relation between semiring of sets and semiring in abstract algebra.

Let a $\mathcal R$ be a family of subsets in $\Omega$ that is closed under finite union and relative complement. We say that $\mathcal R$ is a ring of sets in $\Omega$. Symbolically, for any $A,B\in\...
3
votes
0answers
60 views

Show the equality

Let $(X, \mathcal{A}, \mu)$ space with measure, $\mu(X) = 1$, $\epsilon > 0$ and $f: X \rightarrow [\epsilon,\infty)$ a $\mathcal{A}$-measurable and bounded function, I've tried show $$\lim_{p \...
3
votes
2answers
59 views

Real Analysis, Folland 3.25 Exampes Functions of Bounded Variation

Background Information: Taking $a = -\infty$ and considering the total variation as a function of $b$. To with $F:\mathbb{R}\rightarrow \mathbb{C}$ and $x\in\mathbb{R}$, we define $$T_F(x) = \sup\{\...
4
votes
1answer
32 views

Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
2
votes
1answer
30 views

Sequences of sets, liminf and limsup [closed]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j}^\infty A_i.$$How do I see that$$\...
7
votes
2answers
91 views

Can the Substitution Rule be Interpreted as a “Change of Measure”?

I just started learning measure and rigorous integration theory on my own along side my calculus class and I've noticed that with the substitution rule, you have something that looks like this $$ \int^...
2
votes
1answer
60 views

If $g>0$ is in $L\ln\ln L$, then $\#\{n: g(\theta x)+\cdots+g(\theta^nx)\le t\,g(\theta^nx)\}\le Ct$ when $t\to\infty$

Here are two theorems: For every dynamical system $(X, Σ, m, T )$ and function $f \in L \ln \ln L(X,m)$ (that is, such that $\int |f| \ln^+ \ln^+ |f|\, {\rm d}m$ is finite), $$N^∗f(x)=\sup_{...
5
votes
2answers
76 views

Real Analysis, Folland Theorem 3.18 Differentiation on Euclidean Space

Background Information: A measurable function $f:\mathbb{R}^n\rightarrow \mathbb{C}$ is called locally integrable (w.r.t Lebesgue measure) if $\int_K |f(x)|dx < \infty$ for every bounded ...
3
votes
3answers
29 views

Why does $A_1^\text{c}$ have an infinite number of measurable subsets?

Let $\mathcal{A}$ be a $\sigma$-algebra. Show that if $|\mathcal{A}| = \infty$, then $\mathcal{A}$ is uncountable. We want to construct an infinite sequence of nonempty disjoint measurable sets. ...
5
votes
2answers
37 views

Assuming $\sum_{n = 1}^\infty \int |f_n| < \infty$, properties that follow for integral

How do I see that if $\sum_{n = 1}^\infty \int |f_n| < \infty$, then $\sum_{n = 1}^\infty f(x)$ converges absolutely almost everywhere, is integrable, and its integral is equal to $\sum_{n = 1}^\...
1
vote
0answers
18 views

Equality of random variables measurable w.r.t. different sigma-algebras

I'm stuck trying to prove the following statement: Let $\tau $ be a non-negative random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. We'll consider the following ...
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
57 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
54 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
50 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
37 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
34 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 ...
8
votes
1answer
89 views

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
51 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
31 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
29 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
74 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
122 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 ...