# Tagged Questions

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### Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
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### Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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### Is $\text{Id} = \chi_{\{ |u| \leq k\}} + \chi_{\{|u| > k\}}$ well defined for $u \in L^p(0,T;L^q)$?

Is the decomposition $$\text{Id}(z) = \chi_{\{ |u| \leq k\}}(z) + \chi_{\{|u| > k\}}(z)\tag{1}$$ well defined for $u \in L^p(0,T;L^q(\Omega))$? I guess (1) holds a.e. So the problem is, is the set ...
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### Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
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Let D be the open unit disk in the complex plane, $\mu$ be the arc length measure, and $f, g\in L^{\infty}(\partial D,\mu)$ satisfying the following equations: $$\int_{\partial D} ... 1answer 58 views ### Weak convergence of continuous functions Let X be an LCH space and C_0(X) the set of continuous vanishing functions on X. If C_0(X) is given the structure of a Banach space with the sup-norm, then its weak topology is given by the ... 1answer 51 views ### Simple Functions: Uniform Convergence In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ... 2answers 46 views ### A general question on L^1-space Suppose that \mu and \nu are two (\sigma-finite) measures on a measurable space (\mathbb{X},\mathcal{X}) such that \nu\ll \mu. Is there any result like if E is closed in L^1(\mu), then ... 1answer 18 views ### convergence in measure implies the composition of the sequence of functions and a continuous function also converges in measure Let D be a measureble set in \mathbb{R}^n. Suppose \mu(D)<\infty. Let \phi: D\times \mathbb{R}\to \mathbb{R} be a continuous function such that for almost every x\in D, ... 1answer 35 views ### Dual of L^1 when measure is the counting measure [closed] Let X be an uncountable set, \mu the counting measure on X and \mathcal{M} the \sigma- algebra of countable or co-countable sets. How can I prove that the dual of L^1(\mu|\mathcal{M}) is ... 1answer 17 views ### Lebesgue integral question using du Boise-Reymond lemma This question was inspired a previous question of mine. If we are given that \Omega \subset \mathbb{R}^{n} is open and bounded and$$\int_{\Omega}fv dx = 0 where $f \in C(\Omega)$ and $v \in ... 0answers 40 views ### Compactness hypothesis in Riesz representation theorem Let$X$a compact metric space; I have to identify the dual of the set of continuous functions on$X$,$C(X)^*$. By Riesz representation theorem we have that it can be identified with the space of ... 1answer 49 views ### Counterexample to Marcinkiewicz I have a version of Marcinkiewicz: Let$(X,\mu)$and$(Y,\nu)$be measure spaces and let$1<p_1 \leq \infty$. Suppose that$T$is a mapping from$L^1(X,\mu) + L^{p_1}(X,\mu)$to$\mu$- measurable ... 1answer 24 views ### Pointwise convergence in two variables. I'm not sure about the following (taken from a proof). If$x \rightarrow x_0$and$r\rightarrow r_0$then$\chi_B(r,x) \rightarrow \chi_B(r_0,x_0)$on$\mathbb{R}^n - S(r_o, x_o)$, where$S(r,x) = ...
I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...