# 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.

7,630 questions
Filter by
Sorted by
Tagged with
24 views

### Proving that the function $f(x) := |x|^{\frac{\lambda - n}{p}} (1- \psi(x))$ satisfies two specific properties related with limits and supremums.

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$, where $n \in \mathbb N$ is an arbitrary fixed integer that stands for the dimension of the euclidian space $\mathbb R^n$. In everything ...
• 1,185
1 vote
76 views

### Is $\sin x/x$ Lebesgue integrable over $[0,\infty)$?

Let $f(x)=\frac{\sin x}{x}$ if $x\neq 0$ and $f(x)=0$ if $x=0$. Is $f$ Lebesgue integrable? Also, is $\sin^2x/x^2$ Lebesgue integrable? Note that $\lim_{x\to 0}\frac{\sin x}{x}=1=f(0)$. So $f$ is ...
• 362
25 views

### Use of dominated convergence theorem in Manski (1985)

I'm confused by the use of the dominated convergence theorem (DCT) in Lemma 5 of Manski (1985) (see below). Note that $b = (\tilde{b}_1, \dots, \tilde{b}_{K-1}, b_K).$ Specifically: I presume the ...
• 147
21 views

39 views

• 414
22 views

### integral of the $n$-th power of a function and its distribution

Let $f_1,f_2$ be two measurable functions on $(0,1)$. If thre exists an constant $c>0$ such that $$\int_0^1 f_1^nd m =\int_0^1 f_2^n dm =O((cn)^n) ,$$ then are $f_1$ and $f_2$ equimeasurable?
• 1,348
40 views

• 1,476
1 vote
86 views

### Prove that $f = g$ almost everywhere on $\mathbb{R}$

Let $f$ and $g$ be functions in $L^1(\mathbb{R})$ such that $$\int_E f \, dm = \int_E g \, dm$$ for every measurable subset $E$ of $\mathbb{R}$. Prove that $f = g$ almost everywhere on $\mathbb{R}$. ...
• 111
24 views

• 1,185
1 vote