All Questions
36
questions
0
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97
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Example of Hilbertian norm on the space of radon measures
Assume we are given compact subset $X \subset \mathbb{R}^n$. Consider the space of radon measures $M(X,\mathbb{R})$. I'm trying to find some Hilbert norm on this space either globally or locally. I ...
- 53
0
votes
0
answers
25
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Parameter Dependence Integrals
Let $(X,\mathcal{X},\lambda)$ be a measure space and $(\Theta,d)$ a metric space.
Assume a function $f:X\times\Theta\to \mathbb{R}$ be uniformly bounded, meaning $\sup_{x\in X,\theta\in\Theta}|f(x,\...
- 33
1
vote
1
answer
71
views
Integral of the square of a positive continuous function
Let $F:[0, \infty)\to [0, \infty)$ be a continuous function. It is given that
$$
\int_0^{\infty}f(x) dx<\infty.$$ Determine whether$$
\int_0^{\infty}(f(x))^2 dx<\infty.$$
I think it holds. I ...
- 4,500
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votes
0
answers
17
views
Expression of an integral in terms of lengths of disjoint intervals of a certain set
Let $f$ be a continuous function such that $$f:(0,1)\to[0,\infty).$$
Let
$$S_f(y)=\text{the sum of the length of the disjoint intervals whose union}$$
$$\text{is the set} \{x\in (0,1): f(x)>y\}, \ ...
- 4,500
1
vote
1
answer
42
views
If $f(x):=\mu(\overline{B}(x,r))$ show that $f$ is upper semicontinuous
Let $\mu$ be a Radon measure in $\mathbb R^n$, for fixed $r>0$ define $f(x):=\mu(\overline{B}(x,r))$ show that $f$ is upper semicontinuous.
Like $\overline{B}(x,r)$ is a closed set then $1_{\...
- 311
1
vote
1
answer
65
views
How do I argue that the following Lebesgue integrals are well-defined?
I have the following two iterated integrals:
$\int_{-1}^1\int_{-1}^1 \frac{xy}{(x^2+y^2)^2} dx dy$ and $\int_{-1}^1\int_{-1}^1 \frac{xy}{(x^2+y^2)^2} dy dx$, where $f(x,y)$ is defined on $\mathbb{R}^2\...
- 21
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votes
0
answers
86
views
Dominated Convergence Theorem on an integral
Question
Let $f:[0,\infty)\times [0,1]\to\mathbb{R}$ be a continuous function. Show that $F(t)=\int_{0}^{1} f(t,x) \, dx$ is continuous on $[0,\infty)$.
Ideas
It feels like I'll need to apply the ...
- 477
1
vote
2
answers
674
views
Are $L^p$ functions continuous almost everywhere?
Let $1 \le p < +\infty$ and define the Lebesgue space as the quotient space
$$ L^p(X):= \left\{f\text{: }X\rightarrow \mathbb{R} \mid \int_X \left|f\right|^p d\mu <+\infty\right\}/\sim, $$
where ...
- 1,484
1
vote
1
answer
289
views
$f$ integrable iff set of discontinuities have measure $0$. But why is $\chi_\mathbb{Q}$ not integrable?
I have a theorem stating that if $f:[a,b] \rightarrow \mathbb{R}$ is bounded, then $f$ is Riemann integrable if and only if the set of discontinuities of $f$ has measure 0. Using this you can prove ...
- 391
2
votes
1
answer
264
views
Dominated convergence, and continuity of an integral function
Let $f:[0,1]\times [0,1]\to [0,1]$ be a measurable function with the properties:
$\color{red}{(1)}$ for every $x\in [0,1]$ is $y\mapsto f(x,y)$ continuous
$\color{blue}{(2)}$ for every $y\in [0,1]$ is ...
- 11.2k
4
votes
3
answers
290
views
$\int_0^1 fg\geq 0$ for every non negative, continuous $g$ implies $f\geq 0$ a.e.
I'm trying to solve the following problem.
Let $f$ be an integrable function in $(0,1)$. Suppose that $$\int_0^1fg\geq0$$
for any non negative, continuous $g:(0,1)\to\mathbb{R}$. Prove that $f\geq0$ a....
- 285
1
vote
0
answers
38
views
Non-dense subspace implies non-zero measure
I am reading a proof of the Stone-Weierstrass theorem by De Branges, but I am having trouble understanding the following part.
Let $E$ be a locally compact Hausdorff space and $C_0(E, \mathbb{C})$ the ...
- 103
0
votes
0
answers
38
views
Is the integral of continuous functions over a second argument continuous?
Let $(X,\mathcal A,\mu)$ be a measure space and $Y,Z$ be a normed spaces. Suppose a function $$f: X \times Y \to Z$$ is measurable in its first argument and continuous in its second argument. Moreover,...
- 85
0
votes
1
answer
38
views
Factorization of Characteristic Function
Let $X$ be an open subset of $\mathbb{R}^n$ and for any subset $A\subseteq \mathbb{R}^n$, let $\chi_A$ be the characteristic function of $X$, i.e.:
$$
\chi_X(x):= \begin{cases}
1 & : x \in A\\
0 &...
- 6,779
1
vote
0
answers
32
views
What does it mean for a map taking a measure to an integral to be continuous?
Suppose $\mathcal{M}$ is a set of probability measures. What does it mean to say that a mapping $\mu \rightarrow \int_{\mathbb{R}}f \ d\mu$ is continuous on $\mathcal{M}$? How could one prove that ...
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