# Questions tagged [lebesgue-measure]

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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### On the role of PDFs in the determination of impossible events

Let's consider a random variable $X$ distributed according to a PDF $p(x):\mathbb{R}\mapsto \mathbb{R}_{\geq 0}$. Is it true to say that for any $x_0:p(x_0)=0$ the event $X=x_0$ is impossible? By ...
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### Find all $c\in[3,\infty)$ such that there exists a measure space $(X,S,\mu)$ with $\{\mu(E) : E \in S\} = [0,1]\cup[3,c]$.

I believe that the values of c are 4 and infinity, but I did not understand the explanations you have presented on this site regarding the answer. Do you have better details?
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### Any set of $R^d$ is $G/delta$ or $F/sigma$ [closed]

is this true? We have to try it and we don't know where to start.
1 vote
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### Exercise 8 Chapter 6 Rudin Real and Complex Analysis

I believe I have solved this problem, as it seems relatively straightforward but I am not sure whether the solution has a more 'neat' form in which it can be stated since the conclusion I reached ...
1 vote
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### Yet another property of the Cantor function; a bound on its variation?

Consider the Cantor function $f:[0,1]\to [0,1]$ and an element of the Cantor set $x\in C$, we want to prove that $$f(x+3r)-f(x-3r)\leq 4 (f(x+r)-f(x-r))$$ for all $r>0$ such that the formula makes ...
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### Change of measure for measures on different spaces

For two measures $\nu, \mu$ that are defined on the same measure space with $\nu = \int f d\mu$, it is a well known result, mostly used in the context of the radon nikodym theorem, that for an ...
1 vote
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### Lebesgue measure on $\mathbb{S}^n$ in practice

I want to compute $\int_{\mathbb{S}^n} f(y) d\mu (y)$ with $n=2$, $f$ defined on $\mathbb{S}^n$, and $\mu$ the Lebesgue measure on $\mathbb{S}^n$. How do I evaluate this? Do I just replace $d\mu (y)$ ...
1 vote
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### Is a set zero-measured when the lower dimensional truncated set of which is zero-measured?

I have no clue about this question, so here are some questions that may have answers or counter-examples. I hope this leads to more relevant results. If you know how to deal with any of them, or know ...
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### Integral over random Subset of $\mathbb{R}^2$? [closed]

What is the integral of a random subset $A$: $\int_A f \, d\lambda^2$. I know what it would be if $A$ can be spil up into intervals but what if $A$ is not of some form of intervals?
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Today, in class we were presented this theorem Let $f:[0,1]\mapsto [0,\infty]$ be a bounded, measurable function, prove that $\int_{[0,1]} f\text { }dm=\inf\{\int_{[0,1]}\phi\text{ } > dm\text{ } |... 0 votes 1 answer 53 views ### A question related to a signed measure on the real numbers [closed] I am currently working on the following problem from a past qualifying exam. Let$\nu$be a finite signed measure on$\mathbb{R}.Define a function $$g(x)=\int_{-\infty}^{x}\:d\nu:=\int \chi_{(-\... 0 votes 0 answers 48 views ### Subset V Lebesgue measurable, but not Borel measurable Define the function g: [0, 1] \rightarrow [0, 1] with$$g(y) := \inf\{x \in [0, 1] : f(x) = y\},where f is the Cantor function. Now let V \subset [0, 1] be a set that is not Lebesgue ... 3 votes 0 answers 37 views ### Reference request: almost all level sets of u : \Omega \subset \mathbb{R}^n, u \in L^1(\Omega) have measure zero The following property appears in this other post: If \Omega \subset \mathbb{R}^n is a bounded set and u \in L^1(\Omega), then \mu(\{ x \in \Omega \mid u(x) = t\}) = 0 for almost all t \in \... -1 votes 1 answer 57 views ### Showing equality of first norms We consider a function f \in L_1[0,1], and define g : [0,\infty) \to [0,1] in terms of f, where g(y) = \lambda(|f|^{-1}(y,\infty]) for \lambda the Lebesgue measure, and |f|^{-1}(y,\infty] = ... 0 votes 0 answers 40 views ### There is a function f such that f \in L^{p} and f \notin L^{q} for p,q \geq 1 and p\neq q Problem: For \displaystyle{p \geqslant 1}, consider \begin{align*} f ( x ) = \frac{1}{ \displaystyle{ x^{ \frac{1}{p} } \big( \log^{2}{x} + 1 \big) } } \qquad \text{for all } x \in ( 0 , ... 0 votes 0 answers 22 views ### Show that the image of a homeomorphism involving the Cantor function has a certain measure Currently working on the task of showing the following: Let f be the Cantor function andg: [0,1] \rightarrow [0,1], x \mapsto \frac{f(x)+x}{2}$Then$\lambda(g(C)) = \frac{1}{2}$where$\lambda$... 7 votes 3 answers 173 views ###$\int_0^1 x^n f(x)\,\mathrm{d}x = 0$for all$n$implies$f=0$almost everywhere It has been shown that given$f \in \mathcal C[0,1]$, we have that if$\int_0^1 x^nf(x)\,dx = 0$for all$n \in \mathbb N$, then$f = 0$. I was thinking of generalizing this statement to$L^p$spaces. ... 1 vote 1 answer 38 views ### Verifying the countably additive measure property of$\nu_f$when$\nu_f(E)$is infinite I am studying measure theory and have come across a problem where I am asked to show that a certain function$\nu_f$is a measure. Specifically, let$f \in L^+((X, \mathcal{M}, \mu))$; for any$E \in \...
Let $A\subseteq [0,1]$ be a nowhere dense subset of $[0,1]$ with Lebesgue measure $0$, and let $A'$ be its complement. I want to prove that, for any fixed small $\varepsilon>0$, we can find a ...