# Tagged Questions

For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, 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 is the undergraph definition of Lebesgue integral so rare?

So in Pugh's Real Mathematical Analysis, the initial definition of the Lebesgue integral is as the Lebesgue measure of the undergraph of the function (where the function is nonnegative, with the usual ...
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### 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 ...
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### Definitions of Lebesgue integral

I know the definition, from A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа, of Lebesgue integral of measurable function $f:X\to \mathbb{C}$ on $X,\mu(X)<\infty$ ...
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### Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \|f\|_{wL^p}=\sup_{\gamma>0} \...
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### Show that the set $\{x \in \mathbb{R}| \lim_{n \to \infty} \sin(a_n x) \mbox{ exists}\}$ has zero measure

$a_n$ is a sequence of real numbers such that $a_n \to +\infty$. Show that the set $E = \{x \in \mathbb{R}| \lim_{n \to \infty} \sin(a_n x) \mbox{ exists}\}$ has zero (Lebesgue) measure. The hint for ...
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### I want to understand uniform integrability in terms of Lebesgue integration

According to my Real Analysis textbook, a family $\scr{F}$ of measurable functions on $E$ is said to be uniformly integrable over $E$ provided for each $\epsilon$ $>$ $0$, there is a $\delta$ $>$...
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### $L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
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### Limit of measurable functions in finite measure space

Let $(X,\mathcal{M},\mu)$ be a measure space with $\mu(X)<\infty$. Let $f_n$ be a sequence of measurable real-valued functions such that $f_n$ converges pointwise a.e. to a real-valued function $f$....
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### How to find a measurable but not integrable function or a positive integrable function?

For an arbitrary interval $I$, how can we find a positive on $I$ integrable function? And how does one construct a measurable but not integrable function. If not all measurable functions are ...
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### Intuition behind proof of bounded convergence theorem in Stein-Shakarchi

Theorem 1.4 (Bounded convergence theorem) Suppose that $\{f_n\}$ is a sequence of measurable functions that are all bounded by $M$, are supported on a set $E$ of finite measure, and $f_n(x) \to f(x)$ ...
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### Riemann-Stieltjes integrability criterion

I am currently reading through chapter 11 of Rudin's Principles of Mathematical Analysis, and I'm trying to solve problem 7: Find a necessary and sufficient condition that $f \in \mathfrak R(\alpha)$ ...
Let $\lambda$ denote the Lebesgue measure on the Borel sets of [0,1]. Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous. I know that the Riemann integral $I:=\int_{0}^{1} f(x)dx$ exists. I also know ...
Let $F$ be the power set of $\Bbb{N}$ and consider the measurable space $(\Bbb{N}, F)$. Then what does it mean to take the integral with respect to the measure $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...