Questions tagged [lebesgue-integral]
For questions about integration, where the theory is based on measures. It is almost always used together with the tag [measure-theory], and its aim is to specify questions about integrals, not only properties of the measure.
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Lebesgue integral of L^1 function is differentiable
Let $f\in L^1(\mathbb{R})$, and define the function
$$
F(x)=\int_a^xf(t)dt.
$$
I want to prove that $F$ is almost everywhere differentiable and that $F'(x)=f(x)$ where $F$ is differentiable.
I am ...
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I'm looking for resources that involve concretely taking Lebesgue integral of functions (non-axiomatic and computation focused)
I want to practice finding the Lebesgue integrals of certain functions. My source of inspiration is integrating Dirac delta functions and anything relating to differential equations like Green's ...
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Compactness argument in Browder's proof of the change of variables theorem for multiple integrals
I'm reading Browder's proof of the change of variables theorem for multiple integrals, which begins with the following lemma:
I'm having trouble understanding the highlighted part. That is, I don't ...
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When is convergence in measure useful? [closed]
[Note: By convergence in measure, I mean that $f_n \rightarrow f$ in measure if $\mu(|f_n - f| \geq \epsilon) \rightarrow 0$ for all $\epsilon$. For probabilitists, this is convergence in probability (...
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This expected value has a minimum!
Problem. Let $X$ be a positive, real random variable whose probability density function is bounded by $1$. Prove that $E[X]\geq \frac 12$.
Hi everyone. This problem is essentially saying that the ...
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How do we know the dual pairing between Lp spaces is well defined? [closed]
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
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Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$
Let $\Omega=[0,1]^d$ for some $d\ge 1$, and let $w:\Omega \to (0,\infty)$ be a continuous function.
Is is true that $$\Vert w f \Vert_{L^2(\Omega)}^2+ \left\Vert \frac{1}{w}g \right\Vert_{L^2(\Omega)}^...
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Differentiation under integral signs as done in basic quantum mechanics
In various text books, lectures or lecture notes on basic quantum mechanics, I've seen cases differentiating under integral signs and I am wondering why it is allowed in those situations.
The typical ...
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Excercise on Lebesgue-Integral
I have a question regarding Lebesgue-Integration. I am working with the book by Ziemer (Modern Real Analysis, https://www.math.purdue.edu/~torresm/pubs/Modern-real-analysis.pdf) and I am trying to ...
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Strict inequality of functions only allows to deduce a non-strict inequality of the expected value of said function
In a proof of Jensen's Inequality that I am reading, the following is used:
If for a real valued random variable $X$, we have $X(\omega)<\beta$, then $\mathbb{E}[X]\leq \beta$.
Why can we deduce ...
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Evaluate $\lim_{n \rightarrow \infty} \int_0^{\infty} ne^{-nx} \frac{x^2+1}{x^2+x+1}dx$
I'm trying to evaluate $\lim_{n \rightarrow \infty} \int_0^{\infty} ne^{-nx} \frac{x^2+1}{x^2+x+1 dx}$. I came across this post detailing a solution to it, and wanted to ask if my solution is correct:
...
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Showing $\mu(\{x \in X: |f(x)| > \varepsilon\}) \geq \delta$ [duplicate]
Let $(X, \mathcal{M}, \mu)$ be a measure space with $\mu(X)=1$. For each $1 < p < \infty$, define the set
$$E_p = \left\{f \in L^1(X): \int |f| \, d\mu =1 \hspace{0.2cm} \mathrm{and} \hspace{0....
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A substitution in an integral of Bochner functions
I have given functions $f \in L^2(0,T;L^2(U))$ and $g \in L^2(0,T;H^1(U))$ and a function $H \in C^1([0,T]\times U)$ such that
$$1 \leq H(t,x) \leq 2$$ for a.e. $(t,x)$.
Here, $U=\partial\Omega$ is ...
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Lower bound for mean value of Sobolev function
Assume that for a Sobolev function $f\in W^{2,p}(E)$, for some $p>1$ and $E\subset \mathbb{R}^m$, that the weak derivative $\nabla f$ is essentially bounded in $E$ and that we already know that $|\...
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Can I say $u^{'}(x)=0$ everywhere if $\int_0^1|u^{'}(x)|=0$？
Can I say $u^{'}(x)=0$ everywhere if $\int_0^1|u^{'}(x)|=0$, $u \in C[0,1]$ , $u(0)=u(1)=0$
Background is in Dirichlet problem for unique solution, where $u \in C^2[0,1]$,
suppose two solution exists,...
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