Questions relating to measures, measure spaces, Lebesgue integration and the like.
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3 views
Prove that a function is measurable and its image is a rect line
I´m really stuck with this problem of my homework. I don´t have any idea, how to begin.
Let $f$ be a function, $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x+y)=f(x)+f(y)$ , $\forall ...
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0answers
28 views
question about integral over compact set and bound
Suppose $f(t)$ exists and is finite for a.e. $t \in [0,T]$. Then can I say that $\int_0^T f(t)h(t) \leq K\int_0^T |h(t)|$ for some constant $K$? I know nothing about whether $f$ is continuous, but ...
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1answer
24 views
pointwise convergence implies $L^p$ convergence in this case?
If $f_n(t) \to f(t)$ pointwise and $\int_0^T f(t)$ is finite, does $f_n$ converge to $f$ in $L^p$ for any $p$? I think so, because $f_n$ converges so it's bounded, so one can use DCT. Am I right?
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0answers
15 views
Approximate tangent planes and densities
I'm studying the proof of the following theorem (Simon 'Lectures on Geometric measure theory' Theorem 11.6):
Theorem Let $ M \subset R^{n+k} $ be $ H^n $-measurable (where $ H^n $ is the Haussdorf ...
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1answer
21 views
A clarification about BV functions.
From definition, a locally integrable $ u \in BV(\Omega) $ if its distribution derivative is given by a signed Radon measure. That is there exists $ \mu $ such that for any $ \phi \in ...
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2answers
30 views
If $f \in L_p \cap L_q$, $1 \leq p < r < q < \infty$ then $f \in L_r$.
If $f \in L_p \cap L_q$ and $1 \leq p < r < q < \infty$ then $f \in L_r$.
Proof: Let $E = \{ x \in X: \left\lvert f \right\rvert \leq 1 \}$. Then,
$$\int \left\lvert f \right\rvert^r ...
1
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1answer
26 views
Limit of Lebesgue integral in $L_1([-1,1],m)$
Let $([-1,1],\mathcal{M},m)$ be a measurable space in $[-1,1]$ where $m$ is the Lebesgue measure in $\mathbb{R}$ restricted to $[-1,1]$, and $\mathcal{M}$ is the set of $m^*$-measurable subsets of ...
1
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1answer
26 views
Limit of Lebesgue integrals in $L_1(\mathbb{R},m)$
Let $g\in L_1(\mathbb{R},m)$ bounded function where $m$ is the Lebesgue measure in $\mathbb{R}$. If
$$\lim_{x\to\pm\infty} g(x)=0,$$
show that for all functions $f\in L_1(\mathbb{R},m)$ we have:
...
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0answers
19 views
Converges in measure then a subsequence converges almost everywhere
Let $(X,\mathcal{M},\mu)$ be a measure space. Let $\{f_n\}_{n\in\mathbb{N}}$ a sequence of measurable functions which converge in measure to a function $f$. Prove that exists a subsequence ...
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0answers
29 views
Arbitrarily small open set [duplicate]
Let $E \subset \Re$ be measurable with $\mu(E) > 0$. Show that there
is an $\epsilon > 0$ such that $(-\epsilon, \epsilon) \subset \{x-y \mid x \in E,\, y \in E\}$.
Solution:
...
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1answer
17 views
topology in extended real line
I am having trouble with a simple question :
Consider $\bar{R}$ the extended real line and $ 0 < q < \infty$.
Let $x_n $a sequence in $\bar{R}$ with $x_n \geq 0, \forall \ n $. Suppose that ...
2
votes
2answers
30 views
If $f\in L_p,\varepsilon >0$ then exists simple function $\phi$ where $\|f-\phi\|_p<\varepsilon$
Let $(X,\Omega,\mu)$ be a measure space. If $1 < p < \infty,$ $f\in L_p$ and $\varepsilon >0$ then there exists a simple function $\phi$ where $\|f-\phi\|_p<\varepsilon$.
We did a ...
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0answers
17 views
computing a limit of integrals, maybe using the Convergence dominated theorem [duplicate]
Let's consider a measure space $(X,M,\mu)$ and $f$ a measurable function $f:X\to \mathbb R$ such that $\int f d\mu <\infty$ , for every $\alpha>0$ compute the following limit:
$$
\mathop {\lim ...
4
votes
1answer
48 views
Elements in Product Sigma Algebras
How do I prove that a set is an element of a product $\sigma$-algebra?
We have $(X, m, \mu)$ and $([0,1), \operatorname{Bor}([0,1)), \lambda)$ are measure spaces ($\lambda$ is the Lebesgue measure) ...
4
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0answers
58 views
A combinatorial problem.
Let be $(X, \mathbb{A}, \mu)$ a measure space, a partition of $ X $ is a disjoint family $\xi=\{P_1,\ldots,P_k \}$ of measurable sets such tath $\bigcup P_i=X\pmod0).$
If $\xi=\{P_1,\ldots,P_k ...
