# 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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### $L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
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### Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
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### Does convergence in $L^p$ imply convergence almost everywhere?

If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?
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### Functions that are Riemann integrable but not Lebesgue integrable

I know there are functions which are Riemann integrable but not Lebesgue integrable, for instance, $$\int_{\mathbb{R}} \frac{\sin(x)}{x} \mathrm{d}x$$ Is Riemann integrable and it is easily shown ...
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### How much do we really care about Riemann integration compared to Lebesgue integration?

Let me ask right at the start: what is Riemann integration really used for? As far as I'm aware, we use Lebesgue integration in: probability theory theory of PDE's Fourier transforms and really, ...
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### Why do we restrict the definition of Lebesgue Integrability?

The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.) Why is it we ...
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### What is the intuition behind Chebyshev's Inequality in Measure Theory

Chebyshev's Inequality Let $f$ be a nonnegative measurable function on $E .$ Then for any $\lambda>0$, $$m\{x \in E \mid f(x) \geq \lambda\} \leq \frac{1}{\lambda} \cdot \int_{E} f.$$ What ...
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### Category Theory and Lebesgue Integration.

I'm wondering if there's any Category Theory floating around in the theory of Lebesgue Integration. To avoid things becoming too broad, let's keep this focused on the basics. Here's how I see the ...
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### 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 ...
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### Generalisation of Dominated Convergence Theorem

Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
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### Why is expectation defined by $\int xf(x)dx$?

I recently found out that the expectation of a random variable $X$ in a probability space $(\Omega, \mathcal F, \mathbb P)$, $\mathbb E(X)$, is just the term used in probability theory for the ...
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The problem is the following: Let $a,b,c,d \in \mathbb R$ be given such that $a<b$ and $c<d$. Suppose $f : [a,b] \times [c,d] \to \mathbb R$ is a function such that $\partial_1 f: [a,b] \times [... • 15.9k 22 votes 3 answers 766 views ### Is there a set$A \subset [0,1]$such that$\int_{A \times A^\text{c}} \frac{\mathrm{d} x \, \mathrm{d} y}{\lvert x - y\vert}=\infty$? The above question came up when I was trying to find a counterexample related to this problem. Clearly, the integral of$(x,y) \mapsto \lvert x-y \rvert^{-1}$over$[0,1]^2$is divergent. When ... • 11.7k 22 votes 1 answer 5k views ### A rigorous meaning of "induced measure"? In my readings I often come across terms like "induced measure" or "induced Lebesgue measure". For example: $$\int_{\mathbb{B}^n}u\frac{\partial v}{\partial x_j}\;dx = \int_{\mathbb{S}^{n-1}}uv\... • 4,245 21 votes 3 answers 4k views ### Intuitively understanding Fatou's lemma I learnt Fatou's lemma a while ago. I am able to prove it and use it. I know examples showing that the inequality may be strict. But I don't really have an intuitive way to understand it. Any good ... • 211 21 votes 1 answer 11k views ### If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same? Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample? • 705 21 votes 2 answers 5k views ### A snappy proof of Fatou's lemma I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ... • 2,251 20 votes 3 answers 9k views ### General condition that Riemann and Lebesgue integrals are the same I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ... • 241 20 votes 2 answers 1k views ### Practicality of the Lebesgue integral I am getting pretty frustrated with the Lebesgue integral mainly because it seems highly impractical to calculate anything non-trivial. Whenever I look for a concrete calculation all I see are ... • 500 20 votes 1 answer 3k views ### Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ... • 1,673 19 votes 3 answers 2k views ### How to decide whether Lebesgue integral or Riemann integral? Very often I feel very uncomfortable in dealing with integrals, since I am wondering whether the given integral is meant as a (improper) Riemann integral or Lebesgue integral? For instance, the ... • 645 19 votes 2 answers 3k views ### Various p-adic integrals Here are three (possibly different) definitions of p-adic integrals that I have encountered during my self-studies. First of all, here is what Vladimirov, Volovich and Zelenov write at the beginning ... • 1,586 19 votes 2 answers 2k views ### Show that if the integral of function with compact support on straight line is zero, then f is zero almost everywhere I want to prove that that given f:R^2 \rightarrow R which is continuous with compact support s.t the integral of f for every straight line l is zero (\int f(l(t))\mathrm{d}t=0) then f is ... • 672 18 votes 2 answers 11k views ### Why is the Monotone Convergence Theorem restricted to a nonnegative function sequence? Monotone Convergence Theorem for general measure: Let (X,\Sigma,\mu) be a measure space. Let f_1, f_2, ... be a pointwise non-decreasing sequence of [0, \infty]-valued \Sigma-measurable ... • 3,595 18 votes 3 answers 9k views ### A function that is Lebesgue integrable but not measurable (not absurd obviously) I think: A function f, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain E. However Royden & Fitzpatrick’s book "Real Analysis" (4th ... • 827 18 votes 2 answers 5k views ### Lebesgue integral of a positive function on a set of positive measure Let E be a subset of \Bbb R with positive Lebesgue measure, \lambda(E)>0. Let f be a function from \Bbb R to \Bbb R which is positive on E, that is f(x)>0 for all x\in E. Is ... • 1,173 18 votes 3 answers 12k views ### Integral vanishes on all intervals implies the function is a.e. zero [duplicate] I am having trouble with the following problem: f:\mathbb{R}\to \mathbb{R} is a measurable function such that for all a:$$\int_{[0,a]}f\,dm=0.$$Prove that f=0 for m almost every x (... • 1,906 18 votes 1 answer 3k views ### Difference of differentiation under integral sign between Lebesgue and Riemann Here is a consequence of Lebesgue dominated convergence theorem on differentiation under integral sign. Function f(x, t) is differentiable at x_0 for almost all t \in A, and t \to f(x, t) ... • 333 17 votes 2 answers 2k views ### Does the graph of a measurable function always have zero measure? Question: Let (X,\mathscr{M},\mu) be a measure space and f\colon X \to [0,+\infty[ be a measurable function. If {\rm gr}(f) \doteq \{ (x,f(x)) \mid x \in X \}, then does it always satisfy$$(\mu ... • 77.8k 16 votes 2 answers 8k views ### Measure Convergence Version of Lebesgue Dominated Convergence Theorem I want to prove that LDCT(Lebesgue Dominated Convergence Theorem) continues to hold if I replace the hypothesis$f_n \to f$(convergence pointwise) with$f_n\to f$(convergence in measure): $$\int ... • 7,614 16 votes 4 answers 2k views ### On the horizontal integration of the Lebesgue integral I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ... • 771 16 votes 1 answer 2k views ### Holder's inequality for infinite products In analysis, Holder's inequality says that if we have a sequence p_1, p_2, \ldots, p_n of real numbers in [1,\infty] such that \sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}, and a sequence of ... • 11k 16 votes 2 answers 3k views ### Space \mathcal{L}^p(X, \Sigma, \mu) is separable iff (\Sigma, \rho_\Delta) is separable Let's consider the space \mathcal{L}^p(X, \Sigma, \mu) of all functions f\colon X \to \mathbb{R} (or \mathbb{C}) for which:$$ \int\limits_X|f|^p \mu(dx) < \infty. $$Here X is a metric ... • 3,192 16 votes 1 answer 765 views ### If f is Lebesgue integrable on [0,2] and \int_E fdx=0 for all measurable set E such that m(E)=\pi/2. Prove or disprove that f=0 a.e. Let f be a Lebesgue integrable function on [0,2]. If \int_E fdx=0 for all measurable set E, such that m(E)=\pi/2. Is f=0 a.e. Prove or disprove I could not figure out anything. Can a ... 16 votes 1 answer 4k views ### Is there a general theory of the "improper" Lebesgue integral? I like the Lebesgue integral a lot more than the other alternatives, because of its connections to measure theory; it's a way of thinking about integration that's as "liberated" from the structure of ... • 67.8k 16 votes 1 answer 2k views ### Are continuous functions strongly measurable? Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function f: I \rightarrow X, where I \subset \mathbb{R} is a closed interval and ... 15 votes 2 answers 6k views ### f \in L^1, but f \not\in L^p for all p > 1 "Find an f \in [0,1] such that f \in L^1 but f \not\in L^p for any p > 1." I've thought about doing something like$$f(x) = \frac{1}{x}$$where$|f|^p = \frac{1}{x^p}$doesn't converge ... • 33.9k 15 votes 4 answers 8k views ### Prove that$f$is integrable if and only if$\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$Problem statement: Suppose that$\mu$is a finite measure. Prove that a measurable, non-negative function$f$is integrable if and only if$\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$.... • 2,182 15 votes 3 answers 4k views ### How to prove that$L^p [0,1]$isn't induced by an inner product? for$p\neq 2$I'd like to know how could I prove that$L^p [0,1]$isn't induced by an inner product? (For$p\neq 2$, including$p=\inf$). It is clear to me that I would need to find two functions$f$,$g$in$L^p$... 15 votes 1 answer 6k views ### Suppose$1\le p < r < q < \infty$. Prove that$L^p\cap L^q \subset L^r$. Suppose$1\le p < r < q < \infty$. Prove that$L^p\cap L^q \subset L^r$. So suppose$f\in L^p\cap L^q$. Then both$\int |f|^p d\mu$and$\int|f|^q d\mu$exist. For each$x$in the domain of$...
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Problem: Let $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$ be measure spaces, where $X = Y$ is the interval $[0,1]$, $\mathcal{A} = \mathcal{B}$ is the collection of Borel ...