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.

The idea of Lebesgue integral is the following: we give to a simple non-negative function $\sum_{j=1}^Na_j\chi_{S_j}$, where $a_j\geq 0$ and $S_j>0$ the value $\sum_{j=1}^Na_j\mu(S_j)$. Then we define the integral of a measurable non-negative function as $$\int_X f(x)d\mu(x):=\sup\left\lbrace \int_X g(x)\mathrm{d}\mu(x) \mid 0\leq g\leq f,\ g \text{ simple}\right\rbrace.$$ For a measurable function, write $f=\max(f,0)-\max(-f,0)$ to give a value to $\int_X f(x)\mathrm{d}\mu(x)$.

The major interest is that we can integrate functions which are defined in an arbitrary set, provided we have fixed a $\sigma$-algebra and a measure on it.

When dealing with a function $f\colon[a,b]\longrightarrow\mathbb R$, with $a,b\in\mathbb R$ and $a\lt b$, the Lebesgue integral is more general than the Riemann integral: if a function is Riemann-integrable, then it is Lebesgue-integrable (and the integrals are the same), but there are functions (such as characteristic function $\chi_{[a,b]\cap\mathbb Q}$) which are Lebesgue-integrable, but not Riemann-integrable.

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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 in the case the condition is not satisfied?
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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 for example a function $f(x) = x^2$. How do we…
user957
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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 that its value is $\pi $, nevertheless, it is not…
user438666
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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 exactly is this inequality telling us? Is this saying…
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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 restrict our definition of Lebesgue Integrability to…
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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, anywhere I can think of where integration is used…
Ennar
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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 general set up (missing a few details to keep things…
Shaun
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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 and continuous almost everywhere, which the…
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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? Many thanks! Statement of the theorem: Let $\mu$…
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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 a solution manual for fourth edition? I like…
Gaston Burrull
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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 as a convention so that the rule…
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The Lebesgue integral of an integrable function is continuous

Let $a\in \mathbb{R}$ be fixed, and $f:\mathbb{R}\rightarrow \mathbb{R}$ a Lebesgue integrable function. Define $F:\mathbb{R}\rightarrow \mathbb{R}$ by $$F(x)=\int_a^xf(t)\mathrm{d}t$$ for all $t\in \mathbb{R}$. Prove that $F$ is continuous. Using…
Ducky
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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 that we are set if we show that the integrand is…
Romeo
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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 integrating over a subset of the form $A \times…
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