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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).

Lebesgue measure is the classical notion of length and area to more complicated sets, and its assigns a measure to subsets of $n$-dimensional Euclidean space. Some examples of Lebesgue any closed interval, any cartesian product of intervals, any Borel set, and any countable set of real numbers (which has Lebesgue measure zero).

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Lebesgue measure of an irregular open set using special polygons

I'm now reading Lebesgue Integration on Euclidean Space of Frank Jones,but I don't have an idea of how to pave an irregular open set as in the problem below(Problem4(d) of Chapter2 in Lebegues Integration on Euclidean space,page 30): In the plane…
PhilippeMENG
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Let $(\Bbb R, \mathcal B, m)$ be $\Bbb R$ with Lebesgue measure on Borel sets. Evaluate the measure of a few sets.

Let $(\Bbb R, \mathcal B, m)$ be the measure space $\Bbb R$, with $\mathcal B$ the Borel $\sigma$-algebra and $m$ the Lebesgue measure. Evaluate: $m([−3,5]) =\boxed ?$ $m(\{0,1,2,3\}) =\boxed ?$ $m([0,2] \cup [4,8]) =\boxed ?$
sam
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Lebesgue measure for the graph of a continuous function $f \colon A \subseteq \mathbb{R}^n \to \mathbb{R}^m$

My guess is that I must prove that $A \times \mathbb{R}^m$ has measure zero in $\mathbb{R}^n \times \mathbb{R}^m$. Now, knowing that f is continuous on A, whatever succesion $\{x_n\} \subset \mathbb{R}^n$ s.t. $\{x_n\} \to x \implies f(\{x_n\}) \to…
Eduardo V. Kuri
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Lebesgue measure of sets are equal using Fubini's theorem

Let $f : \mathbb{R}_2 → \mathbb{R}$ be continuous. Let $E$ be lebesgue measurable and $E_f = \{(x,y,z + f(x,y)) : (x,y,z) ∈E\}$. Show that $m_3(E) = m_3(E_f)$ directly from Fubini’s theorem, where $m_3$ is the $3$ dimensional lebesgue measure. I am…
Rene
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Prove that function f(t) = λ(E ∩ [0,t]) is continuous (λ is Lebesgue measure)

Recently I have encountered with this task in my calculus book. Since that I can't get it out of my mind. Task: Let E be measurable subset of $[0,1]$ with positive measure. Prove that $f(t) = \lambda(E \cap [0,t])$ is continuous on $[0,1]$ …
John_doe888_88
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A set has the measure zero

Let $f:[a, b] \to \Bbb R$ be a continuous function such that $f = 0$ almost everywhere; that is, the set $D = \{x \in [a, b]: f(x) \neq 0 \}$ has a measure zero. Prove $f(x) = 0$ for all $x \in [a, b]$. Here is what I was thinking: if $f(x_0) \gt…
bgj123
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Show that $|A| = 0$.

Let $A \subset \mathbb{R}$ such that $\forall x \in \mathbb{R}$ and $r>0$ holds $|A \cap [x,x+r]| \leq rf(r)$ where $\lim_{r \rightarrow 0} f(r) = 0$. Show that $|A| = 0$. I tried to prove the statement when $A$ is bounded, but i failed. Therefore…
user743018
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Is a paraboloid a null set in R3?

Is a circular paraboloid a null set in R3? Intuitively, I think it is not because it has volume in R3. How can I formally prove it?
user832184
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Lebesgue-measure in $\mathbb{R}^2$

For $f : [0,1] \to \mathbb {R}_{\ge 0}$ let $$A = \{ (x,y) \in [0,1] \times \mathbb {R}_{\ge 0}\,|\,0 \le y \le f(x) \}$$ and $$B = \{ (x,y) \in [0,1] \times \mathbb {R}_{\ge 0}\,|\,f(x) \le y \le f(x)+1 \} .$$ Let $\lambda$ be the Lebesgue-measure…
Maria M.
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Requirements of Lebesgue measure theorems

We are looking at the Lebesgue measure on $\mathbb{R}$ and ${(f_n)_{n\in \mathbb {N}}\colon \mathbb {R} \to \overline {\mathbb {R}}}$ with $f_n(x) = 1+\sum _{k=1}^n |x|^k$. Does $(f_n)_{n\in \mathbb {N}}$ fulfill the requirements of a. the dominated…
Maria M.
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Lebesgue measure and Rational enumeration in $[0,1]$

Let $q_1,q_2,\dots$ be an enumeration of all the rationals in $[0,1]$. Define function $f(\omega) = \sum\limits_{n=1}^{\infty} 2^{-n}|\omega-q_n|^{-1/3}$. Prove that $\int_{[0,1]}f(\omega)m(d\omega)<\infty$ where $m$ is the Lebesgue measure. I try…
user387147
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Lebesgue measure of a certain subset in $\mathbb{R}^2$

What is Lebesgue measure of the set $\{(a,b) \in \mathbb{R}^2 \mid a-b \in\mathbb{Q}\}$ in $\mathbb{R}^2$ ? I am guessing this is exactly measure of the diagonal subset, but unable to say rigorously.
dragoboy
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Is the product of a null subset and a subset a null subset?

I'm trying to prove that if A is a null subset of $\mathbb{R^n}$ and B is a subset of $\mathbb{R^m}$, then A x B is a null subset of $\mathbb{R^{n + m}}$. But I'm stuck with this prof, does anyone know how to prove it?
C_Marco
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Does is hold that $f \ge 0$ almost everywhere if $\int_a^b f \varphi dx = 0$ for all $\varphi \in C^\infty_0((a,b); [0,1])$?

Let $(a,b) \subseteq \mathbb{R}$. Suppose $f \in L^1_{\text{loc}}(a,b)$ is real-valued. Suppose that for all $\varphi \in C^\infty_0((a,b); [0,1])$ it holds that $\int_a^b f \varphi dx \ge 0$. Does it hold that $f \ge 0$ almost everywhere on…
JZS
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Suppose $f \in L^1([0,1],\lambda)$ and let $\alpha \in (0,1)$. Assume that $\int_E f=0$ for all$E$ s.t $m(E)=\alpha$ show $f=0$ a.e

Suppose $f \in L^1([0,1],\lambda)$ and let $\alpha \in (0,1)$. Assume that $\int_E f=0$ for all$E$ s.t $m(E)=\alpha$ show $f=0$ a.e. I am not sure how to tackle this problem when $\alpha>1/2$, it is easy when $\alpha<1/2$ by considering the sets…
Sorfosh
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