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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Can I approximate a measurable set with an open set for integration purposes?

I have a Lebesgue measurable function $f:X\rightarrow \mathbb{R}$ where $X\subset\mathbb{R}$. Is there an open set $X^O$ such that \begin{equation*} \int_X f=\int_{X^O} f \end{equation*} and $X^O$ is ...
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1answer
25 views

Help understanding an inequality on Rudin's construction of the Lebesgue measure

I am having trouble understanding an inequality in Theorem 2.20 from "Real and Complex Analysis." Rudin states that if $f\in\operatorname{C}_c(\mathbb{R}^k)$ , $f$ is real, $W$ is an open k-cell ...
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1answer
12 views

example of a prop of measurable functions

We proved in class that a function $f$ is lebesgue measurable if there exists a increasing sequence of simple functions that converge pointwise to $f$. That's OK but how can I build this sequence ...
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17 views

multiplicative inverse of a medible function

I want to prove that if $f$ and $g$ are lebesgue measurable functions then $h$ defined by: $$ h(x):=\frac{f(x)-g(x)}{f(x)+g(x)} \text{ if } f(x)+g(x)\neq 0 \\ 0\text{ if } f(x)+g(x)= 0 $$ ...
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15 views

Lebesgue's measure construction by Halmos.

Could anybody help my to construct Lebesgue measure based on Halmos Measure theory book's. It seems quite incomprehensible to me. What I definied so far: 1)Ring, sigma ring. 2)measure on ring and ...
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2answers
67 views

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<|\mathbb{R}|$, is it true that $X$ is null?

Let $X\subset \mathbb{R}$ Lebesgue measurable, $|X|<2^{\aleph_0}$, is it true that $X$ is null? Of course I am not assuming the Continuum Hypothesis. EDIT: It might be helpful to know that all ...
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2answers
19 views

Convex cone of nonnegative functions in L2 has empty interior

Convex cone $S:=\{f\in L^2(\mathbb{R},\mu):f\geq 0\}$ has empty interior in $L^2(\mathbb{R},\mu)$ when $\mu$ is Lebesgue measure. I wanted to prove it but i have major holes in my knowledge of ...
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Countable unions of Vitali sets…

Let $A \subset \mathbb{R}$ be sets of positive Lebesgue measure. Let $\Gamma$ be a countable dense subgroup of the additive group $\mathbb{R}$. Consider the partition of $\mathbb{R}$ canonically ...
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2answers
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Antiderivative is continuous

The following comes from Bass' book on Real Analysis: (Here $dy$ is Lebesgue measure) Exercise 7.6 Suppose $f:\mathbb{R}\to\mathbb{R}$ is integrable, $a\in \mathbb{R}$, and we define ...
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1answer
21 views

Lebesgue outer measure with open balls

I'm not sure if this has been asked before; if so please redirect me to the appropriate question. The Lebesgue outer measure of $A \subseteq \mathbb{R}^n$ is defined as $$\mu_*(A) = \inf\left\{ ...
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1answer
18 views

Non measurable subset of a positive measure set

I am self-studying measure theory and I have seen this theorem: If $A$ is a set of positive measure, then there exists a subset $D$ of $A$ that is non measurable. I am not sure how to prove it. I ...
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1answer
35 views

Show a subset of $[0,1]^2$ has content $0$

I've been studying Spivak's "Calculus on Manifolds" and I'm thinking about the following question: Let $C \subset [0,1]^2$ be the union of all $\{p/q\} \times [0,1/q]$, where $p/q$ is a rational ...
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0answers
12 views

Interpretation of infinitesimal measure in Lebesgue integration

I have a little trouble understanding the notation of the infinitesimal measure in Lebesgue integration. For example, let's assume I want to compute an volume integral of a function $f: D \rightarrow ...
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26 views

Reference request: Measure theory books using $\omega(\alpha) = |\{f>\alpha\}|$

I am working from Wheeden and Zygmund's Measure and Integral, and they prove theorems such as $\int_E f = -\int_{-\infty}^{+\infty} \alpha d\omega(\alpha)$ where $\omega(a) = |\{x: f(x)>\alpha\}|$ ...
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0answers
13 views

Dominated convergence theorem in case of converge in measure. [duplicate]

I have heard that the dominated convergence theorem hold if almost everywhere convergence is replaced by convergence in measure. I concur if fn converges to f in measure then there exists a ...
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1answer
38 views

monotone convergence theorem( converges in measure)

I have heard that the monotone convergence theorem hold if almost everywhere convergence is replaced by convergence in measure. I concur if fn converge in measure then there exists a subsequence ...
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1answer
23 views

The appropriate translation of Sets of Positive Measure is positive

$A$ and $B$ are two measurable subset of $ \mathbb{R}$, and $m$ is a Lebesgue measure on $\mathbb{R}$, if $m(A)>0$ and $m(B)>0$, then there exist a $ x\in \mathbb{R}$, such that $m(A \bigcap ...
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1answer
34 views

How to use the Bounded convergence Lemma or the Monotone convergence Theorem to calcuate those Lebesgue Integrals?

$\lim\limits_{n\rightarrow\infty}\int\limits_0^\infty n\sin(\frac{x}{n})(x(1+x^2))^{-1}dx$ $\lim\limits_{n\rightarrow\infty}\int\limits_0^1 \frac{1+nx^2}{(1+x^2)^n}dx$ I have tried to show that the ...
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1answer
50 views

Lebesgue measures defined on subspaces of $\Bbb R^n$

For any subspace $V$ of $\Bbb R^n$, we have a special measure $\lambda_V$ which can be described in various ways: Haar measure on $V$, or the measure induced by the metric $V$ inherits from $\Bbb ...
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0answers
31 views

Measure of preimage of nullset under a finite-to-one smooth function is nullset

How can I show rigorously the following which is intuitively clear: Given a smooth function $f: \mathbb R \to \mathbb R$ that is finite-to-one(or which satisfy some other from of 'weak injectivity') ...
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33 views

Error in Hirsch?

Consider the following lemma: In the fragment: Write $\displaystyle X= \bigcup_{1}^{\infty} X_j$ where each $X_j$ is a compact subset of a ball $B$ as above. why is he allowed to do that? If ...
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1answer
19 views

I need a function for the following equality [closed]

I need an example that there exists a measurable non-negative function $f_n:X\to\mathbb{R}$ which uniform converges to $f:X\to\mathbb{R}$, and $\displaystyle\lim_{n\to\infty} \int_X f_nd\mu$ exists, ...
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0answers
37 views

Cauchy's function

An exercise in Real and Complex analysis (exercise 18, page 195,chapter 9) Show that if a function $f$ on the real line $\mathbb{R}$ satisfies \begin{equation*} f(x+y)=f(x)+f(y) ...
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1answer
58 views

A multiple of a characteristic function is the weak limit of a sequence of characteristic functions

Consider $f\in L^1(I,I)$ where $I=[0,1]$ and $ \langle f, g\rangle =\int fg $. For any given $\frac{m}{n}\chi_{A}$ where $\frac{m}{n}$ rational and $A$ an subinterval in $I$, how would I show ...
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1answer
34 views

Sequence of measurable functions $f_n=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$, uniform convergence

For each $n \in \mathbb N$, let $f_n:[0,\infty) \to \mathbb R: f_n(x)=n\mathcal X_{[\frac{1}{n},\frac{2}{n}]}$. Show that there is no $E \subset [0,\infty)$ such that $|E|=0$ and $(f_n)_{n \geq 1}$ ...
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2answers
198 views

Determining a measure through a class of measure preserving functions

Let $\mu$ and $\mu^\prime$ be probability measures over the sigma algebra $\Sigma$ consisting of the Lebesgue measurable subsets of $[0,1]$. Suppose also that $\mu$ and $\mu^\prime$ assign measure $0$ ...
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3answers
424 views

Is the empty set Lebesgue measurable?

I have a quite dumb question. Is the empty set measurable? say with respect to the standard measure. I totally acknowledge intuitive explanations. Thanks,
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23 views

Compute the outer measure of $1+ \frac{1}{n}$

Let us have a fixed interval $I_0=[a,b]$ and let $A$ be a subset of $I_0.$ Compute $$\mu^* \left( \left\{\left( 1+ \frac{1}{n}\right)^n | n \in \mathbb{N} \right\} \right)$$ I've been thinking that ...
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1answer
22 views

$\mu^* \left( \bigcup_{n=1}^{\infty} A_n\right) = 0$

Let us have a fixed interval $I_0=[a,b]$ and let $A$ be a subset of $I_0.$ Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of subsets of $I_0$ s.t $\mu^* (A_n)$ (outer measure) is 0 for all natural $n$. ...
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1answer
35 views

Lebesgue measure. Find $\mu(A)$

If $I_0 = [a,b]$ and $b>a$, let $A \subset I_0$ be a measurable set such that $$\forall p,q \in \mathbb{Q} , p \neq q \rightarrow (\{p\}+A)\cap(\{q\}+A) = \emptyset$$ Then what is $\mu(A)$? ...
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1answer
37 views

Outer measure > $0$?

Let's say we have $A \subset I_0$ as an arbitrary set such that $Int(A) \neq \emptyset$ My question is: is $\mu^* (A)$ always non-negative/positive?
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2answers
39 views

Measure of intersection of set and its translation

I came across an old qualifying exam question: Let $A\subset [0,1)$ be a Lebesgue measurable subset of unit intreval such that $0<\mu(A)<1$. For every $x\in [0,1)$ let $A+x=\{x+y$ mod 1$:y\in ...
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2answers
40 views

Measure theory problem to show a set contains positive interval [duplicate]

Let $E\subset \mathbb{R}$ be a Lebesgue measurable subset of reals such that $\mu(E)>0.$ Consider the set $E+E=\{x+y: x,y\in E\},$ prove that $E+E$ contains an interval of length greater than $0$. ...
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1answer
24 views

Outer measure proof for rational numbers

I saw this problem solved for particular cases like $(0,1)$ but never for general. If $A \subset \mathbb{Q} \cap (a,b)$ and $a<b$ (set of all rational numbers in $(a,b)$) Claim: For every ...
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1answer
31 views

Proof of Fatou-Lebesgue Theorem

Good evening everyone, how can I prove the following inequality? Let $f,f_n\in L(X,\mu)$ on a measure space with $0\leq f_n(x)\leq f(x)$. If $f,f_n$ are $\mu$-almost everywhere in $X$, then is ...
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0answers
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Prove the Pre-image of $T(x,y) = (x-y,y)$ maps Measurable Sets to Measurable Sets

Consider the linear transformation $T(x,y) = (x-y,y)$ on $\mathbb{R^{2n}}$. Prove that that the pre-image of $T$ maps measurable sets to measurable sets. Its clear that T is linear, invertible, and ...
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0answers
27 views

A simple question about Haar measure

$G$ is a locally compact Hausdorff topological (multiplicative) group, $m$ is a (left) Haar measure on $G$. I have known that for any $g\in{G}$, $m(gB)=m(B)$. My question is, for any Borel ...
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0answers
10 views

Image and preimage of sigma algebra and measurable functions

while learning the measure theory one reads a statement like this: Let f be a function from A to B and the corresponding sigma algebra of A. Then the image of the sigma algebra on A is the largest ...
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1answer
32 views

Proving function is measurable

Define $f : [0, 1] → \Bbb R$ by $f(x) = 0$ if $x$ is rational,$1/(d^{1/2})$ if $x$ is irrational and $x = 0.0 . . . 0d . . . $, where $d$ is the first nonzero digit in the decimal expansion of $x$. ...
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1answer
80 views

Can Fatou's lemma and monotone convergence theorem be considered as equivalent?

Can Fatou's lemma and monotone convergence theorem be considered as equivalent?
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106 views

Prove that the Lebesgue integral of $f\chi$ equals to $0$ indicates that $f=0$ a.e.

Suppose $f: [0,1] \to \mathbb{R}$ is bounded, measurable, and $\int_{[0,1]}f \chi_{[0,a)}\, d\mu = 0$ for all $a \in [0,1]$. Prove that $f=0$ a.e. I know that if $\int_{[0,1]}f\, d\mu = 0$, then ...
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1answer
33 views

Proving identities about measurable sets

You are given an interval $[a,b]$ (you can assume WLOG that $a<b$) and you take $A \subset [a,b]$ as a measurable set such that: $$\forall_{c,d\in Q} c\neq d \rightarrow (\{c\}+A) \cap (\{d\}+A))= ...
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1answer
33 views

Computing an outer measure

How do you actually compute an outer measure? I know the definition. It is: $$\mu^*(B):=\inf\left(\sum\limits_{k=1}^n \mu(I_k):B \subset \bigcup\limits_{k=1}^n I_k\right)$$ But how do you use this to ...
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1answer
22 views

Is outer measure always nonnegative?

I know that if you take the measure of the null set, the measure is 0. But say you take a set where the interior of the set is not the empty set. Then is the outer measure of the set positive, and is ...
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20 views

Borel sigma-algebra vs Toplogical space on R

i got two questions on measure theory: The Borel sigma algebra on R is not containing all the toplogical space on R. At the other hand it is generated by all the open sets of the topological space. ...
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1answer
21 views

Question on proof of Borel-Cantelli Lemma

The Borel-Cantelli Lemma: Let $\{E_k\}_{k=1}^\infty$ be a countable collection of measurable sets for which $\sum_{k=1}^\infty m(E_k) < \infty$. Then almost all $x \in \mathbb R$ belong to at most ...
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0answers
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Lebesgue Measurable Sets & Axiom of Determinacy

While reading some logic theory I bumped against the theorem which states that every set of reals is Lebesgue measurable, assuming the axiom of determinacy. To prove this theorem it apparently ...
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1answer
15 views

measurable function and properties of their integrals [closed]

I have to prove that given $(X,M,\mu)$ a measurable space and two measurable functions $\omega_i\colon M \rightarrow [0,\infty]$, $i=1$, $2$, if for all measurable sets E holds $\int_{E} \omega_1 = ...
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1answer
42 views

$Var(F)|_{a}^{b}=\int_{a}^{b}|f|d\alpha.$

Let $[a,b]$ be an interval in $\mathbb R$,and $\alpha :[a,b]\to \mathbb R$ be monotone increasing. Let $f:[a,b]\to \mathbb R$ be integrable, bounded and with respect to $\alpha$. Define $F:[a,b]\to ...
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1answer
52 views

Real Analysis and Lebesgue Measure: Step Functions

For each integer $n$ and $x\in (0,1)$, let $x = 0.k_1k_2k_3...k_nk_{n+1}...$, where $k_i$ is an element of $\{0,...,9\}$ be the decimal expansion of $x$. For such $x$, define $f_n(x) = ...