# Tagged Questions

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### Proof of FTC, continuity part, for Lebesgue integrable functions

The part of the FTC I am interested in says: If $f$ is a Lebesgue-integrable function on $[a,b]$, then $F(x)=\int_a^xf(t)\,dt$ is continuous. This is usually considered a lemma or something for ...
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### McShane vs. Henstock-Kurzweil: Lebesgue integrable

Put in words, is it right to say that the difference of the McShane integral to the Henstock-Kurzweil integral is that the tags are not required to lie within $x_i\leq t_i\leq x_{i+1}$? If so, is ...
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### $L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
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### One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
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Let $X=L^1(\mathbb{R}^n)$ and $T:\mathbb{R} \rightarrow L(X)$, such that $(T(\tau)f)(t):=f(t+\tau)$. The question is: Is $T$ continuous? Well, my idea was the following: $|\tau_1-\tau_2| \le \delta ... 2answers 71 views ### Finding a limit of an integral I am trying to find the following limit. Let$X = [0,\infty)$and$\mathbb B$denote the Borel subsets in$[0,\infty)$,$\lambda$the Lebesgue measure. Let$f_n : [0, \infty) \to \mathbb R$be given ... 1answer 170 views ### Understanding Stokes' theorem Stokes' theorem( here I am only talking about the special$\mathbb{R}^3$case) contains a line integral$\int_{\partial S} \langle f, \tau \rangle ds$. (Actually, I would be confident if somebody ... 2answers 115 views ### Intuition for Lebesgue integration I have started doing Lebesgue integration and I just want to clarify one thing to start with. Very loosely speaking, with Riemann integration we partition the domain into$n$intervals, and then we ... 1answer 113 views ### A dominated convergence theorem applied to$e$number definition I want to show that: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=\sum_{k=0}^\infty \frac{1}{k!}.$$ By the binomial theorem $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = ... 1answer 197 views ### Lebesgue outer measure in dilation invariant? For A \subseteq \mathbb{R}. We know that m^*(A + t) = m^*(A). but Does it follow that$$ m^*(\lambda A) = \lambda m^*(A)$$??? for \lambda > 0. Here m*: P( \mathbb{R}) \to [0, \infty ] is ... 1answer 84 views ### integral over almost sure existing derivatives Let f and g and f-g be real-valued, Lipschitz functions, with Lipschitz constant smaller or equal to 1, on [a,b] whose derivatives are positive and exist only \lambda-almost surely. Does ... 2answers 2k views ### Is Dirichlet function Riemann integrable? "Dirichlet function" is meant to be the characteristic function of rational numbers on [a,b]\subset\mathbb{R}. On one hand, a function on [a,b] is Riemann integrable if and only if it is bounded ... 1answer 108 views ### Integral inequality using positive and negative parts Let f:\mathbb{R} \rightarrow \mathbb{R} be a measurable (with respect to the Lebesgue measure), \pi-periodic function which is Lebesgue integrable over [0,\pi]. Moreover assume that ... 0answers 30 views ### The tightest bound on an integral Consider a polynomial p(x) where p(x)>0 for x\in(0,1) and p(0)=0. Let s(x) be an increasing analytic function such that s(0)=0 and s(1)=1. I am interested to bound the following ... 1answer 147 views ### Prove convergence of improper integral using change of variable. This may be trivial, but I could use some help... Consider a real function f: (0,1) \rightarrow \mathbb{R}, continuous, positive, but not necessarily bounded. Let g: [0,1] \rightarrow [0,1] be a ... 1answer 62 views ### Lebesgue integration— why are contours >t and not =t? I read the Wiki on Lebesgue integration (http://en.wikipedia.org/wiki/Lebesgue_integration) and it says that the integral can be rewritten as the sum of volumes of equal-height contours: The volume ... 3answers 159 views ### How should I calculate the Lebesgue integral of logarithm function from zero to infinity? Does the area under the \ln(x) in (0,+\infty) is measurable? If yes, how can I calculate it? 3answers 190 views ### \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=? If f(x) = \frac{1}{(1+x)\sqrt x} how to find all p > 0 such that$$ \int^{\infty}_0 |f(x)|^p dx < \infty$$The integral is with respect to lebesgue measure. Any solution or hints would ... 1answer 57 views ### The Lebesgue Theory basic Application , get stuck Ok, I am working on a very easy question but I get stuck when I trying to justify my answer. I know that, in order to use Lebesgue's dominated Convergence Theorem, there are two conditions that we ... 2answers 350 views ### Integrability of Derivative of a Continuous Function Let$f$be continuous on$[a,b]$and has finite derivative a.e. on$[a,b]$. Let$f_n(x)=n[f(x+1/n)-f(x)] $s.t.$f_n$be uniformly integrable on$[a,b]$. I want to show :$f'$is Lebesgue integrable. ... 1answer 113 views ### using sup of an unbounded function Is what I'm doing valid if we don't have any information on boundedness of$f$or$f_n$? let$X$be a finite measure space and$\{f_n\}$be a sequence of nonnegative integrable functions,$f_n ...
Let $f$ be a bounded function on a finite interval $[a, b]$ of the real line. If $f$ is Riemann integrable, we denote its Riemann integral by $\mathcal{R}(f , [a, b])$. It is well known that $f$ is ...