For questions about the Lebesgue measure, a measure defined on the Borel or Lebesgue subsets of the real line or $\mathbb R^d$ for some integer $d$. Use it with (tag: measure-theory) tag and (if necessary) with (tag:lebesgue-integral).

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Regularity of Lebesgue outer measure

The terminology in this area is somewhat confusing, my question is how to prove: Given $E \subseteq \mathbb{R}$, there exists a Lebesgue measurable set $A$ such that $E \subseteq A$ and $\lambda^*(E) ...
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28 views

Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$ [on hold]

Could you please help me solving this old prelim problem. Any hints are appreciated
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1answer
50 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
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For $E \subset \mathbb{R}$ and $\epsilon >0$, $\exists$ $(a,b)$ s.t. $\theta(E \cap (a,b)) \geq (1-\epsilon)|b-a|$ ($\theta$ Lesbegue Outer Measure)

In my notes this statement is left unproven. I want to show that for any measurable set $E \subset \mathbb{R}$ with $\theta(E)>0$, there exists an interval $(a,b)$ that covers $E$ arbitrarily ...
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1answer
46 views

What is the 'largest' space of integrable functions which is also a Hilbert space?

It is well known that $L^2(X,\mu)$, the set of functions $f:X \rightarrow \mathbb{C}$ such that $\int_X |f|^2 \text{d} \mu < \infty$, is a Hilbert space. Is there a Hilbert space $H$ such that ...
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31 views

Show $\sup_{y>0}\left|\int_0^\infty \int_t^\infty f(x,y) \cos\left(\dfrac{t}{y}\right)dx\,\,dt\right|<\infty$

Suppose $f$ is Lebesgue measurable on $[0,\infty)\times [0,\infty)$ and $g\in L^1([0,\infty))$. If $|xf(x,y)|\leq g(x)$ for all $y\in [0,\infty)$ prove that $$\sup_{y>0}\left|\int_0^\infty ...
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20 views

Problem with the definition of semi-ring and $\sigma$-sets

I have a problem with a statement I found concerning the definition of semi-ring and that of $\sigma$-set. So, here there is. Assume the definition of a semi-ring $\mathcal{S}$ over a non-empty set ...
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2answers
38 views

prove that $F$ is $\mu\times\mathcal{L}$ measurable where $F(n,x)=\frac{(2n+1)^2\sin((2n+1)x)}{(n(n+1))^2}$

Let $\mu$ be the counting measure on $\mathbb{N}$ and $\mathcal{L}$ be the Lebesgue measure on $[0,\pi]$. Define the function $F$ on $\mathbb{N}\times\mathcal{L}$ by ...
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2answers
42 views

Why is $E[X1_A]=0$ if $P(A)=0$?

I know this is trivial and intuitive, but I'm not able to convince myself rigorously. If $P(A)=0$, why is it true that $E(X1_A)=0$? Every book discards it out as an obvious fact. I tried to prove it ...
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53 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
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1answer
39 views

Slightly Chunky Cantor Sets

I'm familiar with the construction of so-called "fat" Cantor sets (e.g. the Volterra construction), where a Cantor-type construction is used to construct a nowhere-dense set of positive Lebesgue ...
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1answer
38 views

If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$

Problem: If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$ in $L^p$. An official solution I saw for this problem looked very different. Here is my ...
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2answers
26 views

Density function and Integration to $1$

I have a function that's continuous and strictly positive on $\mathbb R$(it's also a density function w.r.t lebesgue to a probability measure), how do I go about defining it if I have the following ...
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3answers
56 views

Is this an identity: $\int_X f \, d\mu=\int_{\mathbb{R}} \mu(f^{-1}(t)) \, dt$?

Let $(X,\mu)$ be a measure space and $f:X \to \mathbb{R}$ an integrable function. Does the following always hold? $$\int_{X}f\,d\mu=\int_{\mathbb{R}}\mu(f^{-1}(t))\,dt$$
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54 views

Probabilistic implications of the existence of non-measurable sets

Measure theory and probability theory are deeply connected through the interpretation of subset measures on the sample space as probabilities of events. A major (and somewhat disturbing) result from ...
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1answer
75 views

Determining dominating function

Using lebesgue dominated convergence, calculate $\displaystyle \lim_{n\to \infty}\int_a^{\infty} n e^{-nx} \cos{x}\, dx$ when $a > 0$ and $a = 0$. I pretty much understand when $a>0$ since ...
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1answer
37 views

Prove that $\int_X f \, d\mu=\int_Y\mu(f^{-1}[t,\infty)) \, d\mathcal{L}(t)$

Let $\mathcal{L}$ be the Lebesgue measure on $Y=[0,\infty)$. Let $(X,\mathfrak{B},\mu)$ be a $\sigma$-finite measure space and let $f$ be a nonnegative $\mu$-measurable function on $X$. Prove that ...
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31 views

L2 norm and L1 norm inequality

In the vector space, we have the following inequality $$ ||x||_2 \leq ||x||_1 $$ where x is a vector. I am wondering that we have similar inequality for function's norm. L1 norm of function f is ...
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12 views

positive integrable part implies downside integrable

Let $A: M\rightarrow GL(d)$ measurable where $(M, \mathcal{B},\mu)$ is a probability space, then are equivalent: $$\log^+\Vert A^{\pm1}(x)\Vert\in L^1(\mu)\Leftrightarrow \log^-\Vert ...
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1answer
27 views

Prelim problem in real analysis (any hints)

Find all constants $K > 0$ for which the following holds: If $(X,\Sigma,\mu)$ is any positive measure space and if $f:X\to \mathbb{R} $ is $\mu$ integrable satisfying $\left|\int_E ...
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17 views

Measurability in the proof of Minkowski's Bound for calculating the Class Number?

I recently looked at the proof of Minkowski's Bound given in Number Rings, and it appeared to implicitly rely on the interesting (and somewhat non-trivial) fact that convex subsets of $\mathbb{R}^n$ ...
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1answer
42 views

If a sequence $f_n$ is bounded in $L^2$ and converges to zero a.e., then $f_n\to 0$ in $L^p$ for $0<p<2$

Let $M>0$, $\{f_n\}\subset L^2([0,1])$ such that $\int_0^1 |f_n|^2 dm\leq M$ and $f_n(x)\to 0$ as $n\to\infty$ almost everywhere, $m$ is Lebesgue measure. Show that for all $0<p<2$, ...
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1answer
42 views

Distribution of transformed random variables

We have that f is a density w.r.t the lebesgue measure $m$ for a probability measure on $\mathbb{R}$, that f is continuous and strictly positive. X and Y are to random variables s.t. the distribution ...
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2answers
32 views

How can I show that the “binary digit maps” $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ for ...
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39 views

Lebesgue measure is separable?

I would like to better understand the following definition: $(M, \mathcal {A}, \mu) $ a probability space is separable if there exists a countable family $ \mathcal {E} \subset \mathcal {A} $ such ...
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1answer
83 views

$\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$ implies $\lim_{n \to \infty} \int_B f_n \, d\mu = \int_B f \, d\mu$ for $B \subseteq X$

I'm having trouble with the following problem. Let $(X, \mathcal{M},\mu)$ be a measure space, where $X = [a,b] \subset \mathbb{R}$ is a closed and bounded interval and $\mu$ is the Lebesgue measure. ...
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1answer
46 views

Find a function in $L^p(\mathbb{R})$ only for $p=4$ [duplicate]

I'm having trouble with this problem from an old analysis qual: Find a function $f$ such that for $p\in (1,\infty)$, $f$ is in $L^p(\mathbb{R})$ only when $p=4$.
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1answer
36 views

Convergence of series by using counting measure

Problem; Let $\{a_n\}$ and $\{r_n\}$ be two sequences of real numbers such that $\displaystyle\sum_{n\geq 1} |a_n|<\infty$. Prove that $$\sum_{n\geq 1} \frac{a_n}{\sqrt{|x-r_n|}}$$ converges ...
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32 views

What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion?

Let $S= \{x \in [0,1 ]: \lim_{n \to \infty } \frac {1 } {n } \sum _{i=1 } ^n d _i (x) = 2/3 \} $, where $d _i(x) $ gives the ith digit of the infinite base-2 expansion of $x $. What is the Lebesgue ...
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1answer
23 views

Understanding a proof using Caratheodory's criterion

I'm reading a proof on proving that countable unions of measurable sets are again measurable. The proof is as follows Let $T$ be a test set. $E_1\cup E_2$ is measurable if Caratheodory's ...
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2answers
21 views

Open/Closed set in Lebesgue measuer

Let $E\subset \mathbb{R}$ such that $\lambda(E)=0$ (the Lebesgue measure on $\mathbb{R}$). Can $E$ be open? Must it be closed?
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1answer
20 views

Countable union of measurable sets is measurable?

Is a countable union of measurable sets measurable? If the sum of measures of those measurable sets is finite, then their union is also measurable. But if the sum of measures of measurable sets is ...
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1answer
27 views

Applications of Singular Functions

For our purposes here, a singular function is a continuous function such that the part which is absolutely continuous with respect to Lebesgue measure is zero. For example, the Cantor function or ...
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Prove that G is absolutely continuous and not monotone on any subinterval of $[0,1]$

Let $A\subset [0,1]$ be a Borel set such that $0<m(A\cap I)<m(I)$ for any subinterval $I$ of $[0,1]$. Let $G(x)=m([0,x]\cap A)-m([0,x]\setminus A)$. Prove that G is absolutely continuous and not ...
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1answer
31 views

Proving that all Lebesgue measurable characters are continuous.

Given that a character on $\mathbb{R}^1$ is defined as a function (complex-valued function) such that $|\phi(t)|=1$ and $\phi(s+t)=\phi(s)\phi(t)$ for all $s,t\in\mathbb{R}$, how would one go about ...
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1answer
21 views

Lebesgue integrable discontinuity points

If a function is Lebesgue integrable, is it possible that it has as set of discontinuity points measure bigger than zero?
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46 views

How to use dominated convergence theorem?

How to use dominated convergence theorem to compute $$lim_{n\rightarrow \infty}\int_0^1\frac{1+nx^2}{(1+x^2)^n}$$ So far I have only done $\frac{1+nx^2}{(1+x^2)^n}\le\frac{1+nx^2}{(1+x^2)}$, I don't ...
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1answer
57 views

A function constant almost everywhere

Q: Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is measurable with respect to Lebesgue measure and $f(x)=f(x+1)=f(x+\pi)$ for almost every $x$. Prove that $f$ is constant almost everywhere. Proof ...
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1answer
49 views

Increasing functions on $\mathbb{R}$

If $F$ is increasing on $\mathbb{R}$ then show that $F(b)-F(a)\geq \int_a^b F'(t)dt$. My work: Since $F$ is increasing on $\mathbb{R}$, $F'$ exists a.e. on $\mathbb{R}$. So $F'$ is integrable on ...
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1answer
32 views

Show that: $ X \ {\text{is}}\ \mu{\text{-integrable}} \implies \sum_{k=1}^\infty\mu(\{\mid X\mid ≥ k\}) < \infty$

Assignment: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $X: \Omega \rightarrow \bar{\mathbb{R}}$ a $\mathfrak{A}$-$\bar{\mathfrak{B}}$-measurable function. Show that: $$ X \ ...
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2answers
32 views

Show that: $\int_{\Omega} X d\mu = 0 \iff \mu(\{\omega\in\Omega\mid X(w)>0\})=0$

Assignment: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $X: \Omega \rightarrow \bar{\mathbb{R}}$ a non-negative $\mathfrak{A}$-$\bar{\mathfrak{B}}$-measurable function. Show that: ...
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1answer
45 views

An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$

I am thinking about this problem: Let $f\in L^1 [0,1]$ to be a nonnegative function satisfied: $$\int_{E} f dm\leq \sqrt{m(E)}$$ for every measurable set $E\subset [0,1]$, Prove that $f\in ...
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1answer
27 views

Show an integrable function a.e. 0

$f\in L^1[0,1]$, there is a constant $0<c<1$, suppose for any measurable set $A\in [0,1]$ with $m(A)=c$, we have $$\int_A f =0$$ Prove that $f=0$ a.e. I know how to prove this when $f$ is ...
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0answers
31 views

proof that the lebesgue measure of a subspace of lower dimension is 0.

I asked this question yesterday Lebesgue measure of a subspace of lower dimension is 0, Matt S answer didn't really convince me and after trying to find a solution without using determinants and the ...
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0answers
39 views

Refinement of Lebesgue Decomposition Theorem

On Wikipedia, a "refinement" of the Lebesgue decomposition theorem is given, and it is also given as problems in Stein and Shakarchi and Bruckner and Thomson. Can someone provide a comprehensive proof ...
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1answer
36 views

A set with positive measure has a point for which every open interval around it has positive measure

Let $\mu$ be a measure on $(\mathbb{R},\mathcal{B})$. Let $B \in \mathcal{B}$ be such that $B \in (-\infty,\infty) $ and $\mu(B) >0$. Give a proof or else give a counter example to the assertion: ...
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2answers
37 views

Lebesgue measure of a subspace of lower dimension is 0

I'm currently reading Rudin's book Real and Complex analysis. In page 52 he says To prove (e) let $T:R^k\to R^k$ be linear. If the range of T is a subspace Y of lower dimension then $m(Y)=0$. I ...
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2answers
28 views

If $f_n \rightarrow 0$ and $\int \sup f_1, … , f_n \leq M$ then $\int f_n \rightarrow 0$

Let $f_n$ be a sequence of nonnegative measurable functions which converge to $0$. If there exists an $M$ such that $$\int \sup f_1, ... , f_n \leq M$$ for all $n$, then $\lim \int f_n = 0$. Could ...
4
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2answers
41 views

Prove the following inequality: $\int_{(a,b)}f\ d\lambda\cdot\int_{(a,b)}\frac{1}{f}d\lambda≥(b-a)^2$

Assignment: Let $-\infty < a < b < \infty$ and $f: (a,b) \rightarrow (0,\infty)$ be measurable, such that $f$ and $\frac{1}{f}$ are Lebesgue integrable. Prove the following inequality: ...
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1answer
21 views

Existence everywhere of integrand in Fubini's theorem

Let $f\in L(A,\mu_x\otimes\mu_y)$ be a summable function on $A\subset X\times Y$ where $(X\times Y,\mu_x\otimes\mu_y)$ is the product of measure spaces $(X,\mu_x)$ and $(Y,\mu_y)$. Then Fubini's ...