# 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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### Without Lebesgue

Everyone knows following problem. Let $f$ be positive function on $[0,1]$ and there exist $I = \int_{0}^{1}f(x)dx$. Prove that $I>0$. (recall that there are only two cases: $I=0$ or $I>0$. NOT ...
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### Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
Let $G$ be a compact group. Let $\mu'$ and $\mu$ be the Haar measure on $G \times G$ and $G$, respectively, and further such that $\mu'(G \times G) = 1$ and $\mu(G)=1$. Does it follow that $\mu' = \mu ... 0answers 31 views ### Basic facts related to Haar measure I have a compact group$G$and continuous functions$f_1, f_2$from$G$to$\mathbb{C}$and$g: \mathbb{R} \rightarrow \mathbb{C}. I have two questions related to Haar meausure. Is it true that $$... 2answers 44 views ### p-norm of a function Let f\in L^1(\mu)\cap L^\infty(\mu). I have proved for any 1<p<\infty, f\in L^p(\mu), w(p)=||f||_p is continuous w.r.t. p, and \lim_{p\to \infty}||f||_p=||f||_\infty. Is w(p) ... 1answer 48 views ### Integration using Lebesgue dominated convergence theorem This is an old comp question I'm working on.$$\lim_{n\to\infty}\int_{[0,1]}\frac{d\lambda}{x^\frac{1}{n}(1+\frac{x}{n})^n}I am having trouble finding a dominating function. Thinking about the ... 1answer 34 views ### Product measure and integrals of simple functions Let (\Omega_1 , \mathcal{X}, \mu) and (\Omega_2 , \mathcal{Y}, \nu) be two \sigma-finite measure spaces, and let \mu \times \nu be product measure on the \sigma-algebra \mathcal{X} \times ... 3answers 46 views ### Positive integral everywhere implies positive function a.e I would like to get feedback on my demonstration of this simple statement : Let f be an integrable function on the measure space (X,S,\mu). \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ... 1answer 56 views ### Functions with every point being a Lebesgue point For a locally integrable function f a point x is a Lebesgue point if the integral averages of deviations from f(x) over balls centered at x converge to 0 as the balls shrink to the point. ... 4answers 98 views ### Are null sets necessarily closed? Hi everyone: Is a null set of \mathbb{R}^n, (n>0), necessarily closed? Give a counter example. Thanks for your reply. 1answer 67 views ### An amazing inequality of the integration of two functions. Let f:[0,1]\longrightarrow\mathbb{R} be measurable and g\in L^1[0,1] such that for all t>0, \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$Prove that for 1<p<2,$$ ... 1answer 26 views ### Integrability of the logarithm wrt a finite Borel measure I have a finite Borel measured\phi$on$(0,1)$, i.e.$\int_0^1 d\phi(x) < \infty$. Is it also true that$\int_0^1 \log (x) d\phi(x) < \infty$? The function$\log$is integrable at 0, so ... 1answer 116 views ### When is$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$An elementary question on Riemann - Integration: Under what conditions on$f$is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If$f$is bounded in$[a,b]$, then this is ... 1answer 24 views ### Integrability in a domain of$\mathbb{R}^{N}$Hi everyone: Let$f$be a function defined on an open set of$\mathbb{R}^{N}(N\geq1)$. Is there any difference between the following two statements? 1)$f$is locally integrable 2)$f$admits a ... 0answers 51 views ### Is$\sigma$-finiteness really a necessary condition for this problem? Question: Let$(X, \mathcal A, \mu)$be a measure space and suppose$\mu$is$\sigma$-finite. Suppose$f$is integrable. Prove that given any$\varepsilon$, there exists a$\delta >0$such that ... 1answer 27 views ### an argument that strengthen Lusin's theorem Let$f$be a measurable function on a subset$E$of$\mathbb{R}^n$. Lusin's theorem states that for any$\epsilon>0$, there exists a measurable subset$F$such that$F$open in$E$, ... 1answer 68 views ### Why is$fg$integrable w.r.t. a probability measure if$f,g$are Lebesgue integrable? In one of the proofs, my text mentions that if$f,g$are Lebesgue integrable then$fg$is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ... 2answers 71 views ### Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net$ (f_{\alpha})_{\alpha \in A} $of measurable functions? Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net$ (f_{\alpha})_{\alpha \in A} $of measurable functions defined on a measure space$ (\Omega,\Sigma,\mu) $, where the index ... 0answers 57 views ### An inequality between integrals of series of characteristic functions of cubes Let$1\leq p<\infty$. Prove that there exists$C>0$such that $$\left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ... 1answer 21 views ### Probably simple application of Holder or Minkowski for integrals This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose U and V are integrable over measure space ... 0answers 20 views ### Change of variable for Lebesegue Integral Let G be an absolutely continuous function, G:[a,b] \rightarrow [c,d] and f \geq 0 a Lebesegue measurable function in [c,d]. I managed to prove that if f is just Borel measurable it holds ... 0answers 49 views ### Hardy Littlewood maximal function and integral comparison. Define the Hardy Littlewood maximal function$$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$For given x_i,r_i,a_i, first I have shown that ... 1answer 18 views ### Fine Print of Egorov's Theorem The Egorov's theorem in my textbook requires that the function to be define on a set with finite measure. Why is this necessary, please? Thank you! 1answer 40 views ### Two questions on Fatou's Lemma While reading the following paragraph from Real Analysis by Stein (I hope this does not breach any copyright; if so, I have to type it out), two questions occurred to me. In the proof of Fauto's ... 1answer 32 views ### L^p norm of a measurable function is bounded by its operation on step functions Let 1\leq p<\infty, 1/p+1/q=1. Let f be a measurable function on [0,1] such that for all step functions g on [0,1]$$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$Prove that ... 1answer 46 views ### a condition given by step functions implies the condition holds for L^q space Let 1\leq p<\infty, 1/p+1/q=1. Let f be a measurable function on [0,1] such that for all step functions g on [0,1],$$ |\int_0^1 fg d\mu|\leq ||g||_q. $$Prove ||f||_p\leq 1. How ... 1answer 15 views ### comparing mean values of a positive functions Suppose that D is a bounded open set in \mathbb{R}^{n}, and A\subset B\subset D measurable sets. Let f:D\rightarrow [0.+\infty) be a measurable function (or even locally integrable) and ... 1answer 28 views ### Integration with respect to conditional measure? Let (X_n) be a Markov chain. For i\in S my text defines$$N_i:=\sum_{n=0}^\infty \mathbf 1_{\{ X_n=i \}}$$and then, as a part of a larger proof, claims that$$\mathbb E_i(N_i)=\sum_{n=0}^\infty ... 1answer 37 views ### Limit of average of real function I need some hints regarding this exercice. if$f : [0, \infty)\rightarrow \mathbb{R}$is a measurable function s.t$\lim_{x\rightarrow \infty} f(x) = a, prove : \begin{align} \lim_{x\rightarrow ... 1answer 71 views ### Spivak's “Calculus in Manifolds” problems I have some troubles with this problems. Problem 1.18: IfA \subset [0,1]$is the union of open intervals$(a_i,b_i)$such that every rational number of$(0,1)$is contained in$(a_i,b_i)$, for ... 2answers 23 views ### Exercise on abstract integration Let$f_n$be a sequence of nonnegative functions defined on$\mathbb{R}^N$such that$f_n \rightarrow f $almost everywhere on$\mathbb{R}^Nand such that $$\int_{\mathbb{R}^N} f_n \rightarrow ... 1answer 24 views ### the area of the image under a specific holomorphic function of the unit disk Let f(z)=z^3+\frac{z^2}{2}. Let D be the unit disk in \mathbb{C}. How to compute$$ Area(f(D))? In the case that f:D\to \mathbb{C} is injective, \begin{align*} Area(f(D))&= \int_D ... 3answers 42 views ### Question on integral, notation and Nikodym derivative I have a very general question for those measure theoric, real analysis guy out there . I am very confused by the concept of Nikodym derivative. If v << \mu, we can find a non negative ... 1answer 36 views ### Reversing limits in Lebesgue integration I know that reversing limits of Riemann integration is possible by putting minus sign. My question is that there is a similar result for Lebesgue integral as well. For example, ... 2answers 66 views ### solution of an integral equation in measurable functions Let\phi(t)$be a positive continuous function on$[0,\infty)$and$f(t,x)$be a continuous function of two variables such that $$|f(t,x)|\leq \phi(t)|x|.$$ Suppose ... 1answer 33 views ### inequalities concerning integration and measure Let$f$be a non-negative function on$\mathbb{R}^n$such that$\int_{\mathbb{R}^n} f=1$. Let$p\in(0,1)$. Let$E$be any measurable subset of$\mathbb{R}^n$. Prove that $$\int _E f^p\leq ... 1answer 51 views ### Simple Functions: Uniform Convergence In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ... 2answers 57 views ### \lim_{A \to \infty} \int_0^{A} \int_0^{\infty} \sin(x) e^{-xt}dtdx I would like to compute the following integral:$$\lim_{A \to \infty} \int_0^{A} \int_0^{\infty} \sin(x) e^{-xt}dtdx \qquad (1)$$I would like to swap the order of integration because then the ... 1answer 33 views ### counterexample of Riemann-Lebesgue lemma for non-Borel functions Let f:\mathbb{R}\longrightarrow \mathbb{R} be a Borel measurable function. Then$$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$I obtain this result by showing that it is ... 1answer 17 views ### Lebesgue integral question using du Boise-Reymond lemma This question was inspired a previous question of mine. If we are given that \Omega \subset \mathbb{R}^{n} is open and bounded and$$\int_{\Omega}fv dx = 0$$where$f \in C(\Omega)$and$v \in ...
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 ...