# 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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### 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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### 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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### 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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### 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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### 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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### Is there a fundamental reason that $\int_b^a = -\int_a^b$

Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
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### If $\int_{\mathbb R^2} \frac{\vert f(x)-f(y)\vert}{\vert x-y\vert^2}dxdy<+\infty$ then $f$ is a.e. constant

Let $f \in L^1(\mathbb R)$. If $$\int_\mathbb R \int_\mathbb R \frac{\vert f(x)-f(y)\vert}{\vert x-y\vert^2}dxdy<+\infty$$ then $f$ is a.e. constant. I do not know how to begin. I thought ...
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### 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 ...
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### 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 ...
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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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### Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
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### If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere

I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some ...
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### 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 ...
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### 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 $t \to f(x, t)$ ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
557 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 ...
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### When is $\int_0^1 \int_0^1 \frac{f(x) - f(y)}{x-y} \, \text{d} x \, \text{d} y = 2 \int_0^1 f(t) \log\left(\frac{t}{1-t}\right) \, \mathrm{d} t$?

Double integrals of this type sometimes appear when using differentiation under the integral sign with respect to two variables. Therefore, I am interested in reducing them to (simpler) single ...
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### 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 ...
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Let $(X, M, m)$ be an arbitrary measure space. Let $f_n, f \in L^1_m(X)$. Assume that $$\int_X |f_n - f| \, dm \leq \frac{1}{n^2} \text{ for all }n \geq 1.$$ Then I want to show that $f_n \... 2answers 10k views ### Bounded convergence theorem Suppose that$f_n$is a sequence of measurable functions that are all bounded by M, supported on a set E of finite measure, and$f_n(x)\to f(x)$a.e. x as$n\to \infty$. Then f is measurable, bounded, ... 1answer 218 views ### Convergence of a series of translations of a Lebesgue integrable function Let$f: \mathbb{R} \rightarrow \mathbb{R}$be a Lebesgue integrable function. Prove that $$\sum_{n=1}^{\infty} \frac{f(x-\sqrt{n})}{\sqrt{n}}$$ converges almost for every$x \in \mathbb{R}$. My ... 0answers 345 views ###$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$implies$u\in L^{6/5}(\Omega)$Let$d=3$and$\Omega\subset \mathbb R^d$is a bounded Lipschitz domain and$u$is a measurable function. A sufficient condition for the integral$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^...
I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ \int_*f:=\left\{\int h ; h \...
### What does it mean to be an $L^1$ function?
I am struggling to understand what the space $L^1$ is, and what it means for a function to be $L^1$. A friend told me that a function $f$ is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if \$(...