All Questions
Tagged with integration measure-theory
307
questions
36
votes
3
answers
12k
views
If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?
Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ ...
- 11.8k
22
votes
6
answers
20k
views
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 ...
- 2,259
58
votes
5
answers
28k
views
Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure
Is there a generic change of variables formula for a measure theoretic integral that does not use the Lebesgue measure? Specifically, most references that I can find give a change of variables ...
- 581
12
votes
4
answers
3k
views
Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
Let $({\Bbb R},{\mathcal A},\mu)$ be the measure space where ${\mu}$ is the Lebesgue measure. Assume that $\int_{\Bbb R}|f|^pd\mu<\infty$ ($p\geq1$). There is an exercise for proving that
$$\...
user9464
5
votes
1
answer
1k
views
surface measure and Gauss-Green theorem proof
The famous Gauss-Green theorem states the follows.
Let $\Omega$ be a bounded open set of $\mathbb{R}^n$and $\partial \Omega$ be $C^1$ boundary. Then for $f,g \in C^1(\overline{\Omega}),$ the ...
- 1,174
71
votes
2
answers
29k
views
Integration with respect to counting measure.
I am having trouble computing integration w.r.t. counting measure. Let $(\mathbb{N},\scr{P}(\mathbb{N}),\mu)$ be a measure space where $\mu$ is counting measure. Let $f:\mathbb{N}\rightarrow{\mathbb{R}...
user54992
34
votes
4
answers
5k
views
Is there any difference between the notations $\int f(x)d\mu(x)$ and $\int f(x) \mu(dx)$?
Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$$\int f(x)d\mu(x)$$
and the notation
$$\int f(x) \mu(dx)$$?
- 519
14
votes
3
answers
13k
views
Prove Minkowski's Inequality for Integrals
I am interested in proving the following claims :
Suppose that ($X$, $\mathcal{M}$, $\mu$) and ($Y$, $\mathcal{N}$, $\nu$) are $\sigma$-finite measure spaces, and let $f$ be an ($\mathcal{M} \otimes \...
- 141
9
votes
1
answer
2k
views
Prove that $\int_{E}f =\lim \int_{E}f_{n}$
I'm doing exercise in Real Analysis of Folland, and got stuck on this problem. I try to use Fatou lemma but can't come to the conclusion. Can anyone help me. I really appreciate.
Consider a ...
- 4,845
7
votes
2
answers
3k
views
Jordan measure zero discontinuities a necessary condition for integrability
The following theorem is well known:
Theorem: A function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero.
Now if we change ...
- 3,586
6
votes
3
answers
942
views
Proof of the identity $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$ for $\alpha\in (0,2)$.
Let $0<\alpha<2.$ Looking for a proof for the following: $$\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}.$$
Any ideas?
- 1,070
8
votes
2
answers
2k
views
Summing over General Functions of Primes and an Application to Prime $\zeta$ Function
Along the lines of thought given here, is it in general possible to substitute a summation over a function $f$ of primes like the following:
$$
\sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1}
$$
and ...
- 18.6k
50
votes
8
answers
6k
views
Why do we restrict the definition of Lebesgue Integrability?
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we ...
- 2,924
17
votes
3
answers
1k
views
"Proof" all integrals are $0$
Given some definite integral over $[a,b]$ make the substitution $ u = x(x-a-b)$. The integral is then transformed to an integral over $[-ab,-ab]$ which is zero. What is wrong with this reasoning? When ...
- 4,284
6
votes
2
answers
297
views
Working with infinitesimals of the form d(f(x)), for example d(ax), and relating them to dx (integration, delta function)
I am trying to get a better understanding on how we can manipulate the infinitesimal dx in an integral $$\int f(x) dx$$
I have come across the following
$$ d(\cos (x)) = -\sin(x) dx$$
Therefore
$$\int^...
- 443