All Questions
Tagged with measure-theory integration
2,737
questions
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8
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Strict inequality of functions only allows to deduce a non-strict inequality of the expected value of said function
In a proof of Jensen's inequality that I am reading, the following is used: If for a real valued random variable $X$, we have $X(\omega)<\beta$, then $\mathbb{E}[X]\leq \beta$. Why can we deduce ...
- 353
-3
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1
answer
63
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Explanation for $m(O-E) = m(O) - m(E)$ [closed]
I am self studying Measure Theory from G. de Barra. Please can anyone give a detailed explanation for the following:
$$m(O-E) = m(O) - m(E)$$ where E is a subset of O, it's the second line after the ...
1
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64
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What's the definition of a line integral on a possibly disconnected curve?
I'm trying to understand this paper, and I see this integral (page 2):
$$
\int_{B\ \cap\ \mathcal{C}} (1 - y)dy,
$$
where $\mathcal{C}$ is the curve given by $x = ye^{1-y}$ and $B$ can be any ...
- 1,929
1
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1
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35
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Prove: If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.
I need to prove the following result:
Let $(X,\mathscr{A},\mu)$ be a measure space. If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.
Here is my ...
- 2,473
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31
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Is there a simple proof that the Laplace transform of a signed measure $\mu$ on $\mathbb R$ uniquely determines $\mu$? [duplicate]
A Laplace transform of a signed measure $\mu$ on $(\mathbb R, \mathcal B(\mathbb R))$ is defined by
$$ f_\mu(s) = \int_{\mathbb R} e^{-\lambda t} d \mu(t), \qquad \forall s \in \mathbb R. $$
I know ...
1
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0
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58
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Prove: $F:M(X,\mathscr{A},\mathbb{R})\to\mathbb{R}$ defined by $F(\mu)=\int fd\mu$ is a linear functional. [closed]
I need to prove the following result:
Remark 4.30$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Let $B(X,\mathscr{A},\mathbb{R})$ be ...
- 2,473
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1
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38
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Integration with spherically symmetric measure in $\mathbb R^d$
Let $\mu$ be a finite spherically symmetric measure over $\mathbb R^d$, so that $\mu(TB) = \mu(B)$ for all orthogonal transformations $T: \mathbb R^d \rightarrow \mathbb R^d$ and Borel set $B$. Let $g:...
- 581
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64
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Find the limit $\lim_{k \rightarrow \infty } \int_0^k x^{-k} e^{x^2/k^2}\sin(x/k)~\mathrm dx$
Compute the limit$$\lim_{k \rightarrow \infty } \int_0^k x^{-k} e^{x^2/k^2}\sin(x/k)~\mathrm dx.$$
Completely stuck with this one. Some convergence theorem is obviously needed, but can't figure out ...
- 17
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1
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59
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Estimating integrals and measures over Hilbert space using finite dimensional projections
Let $H$ be a separable Hilbert space with orthonormal basis $\{e_k\}_{k \in \mathbb{N}}$. Let $P_n$ be the projection onto the $n$-dimensional subspace of $H$:
$$P_n x = \sum_{i=1}^n \langle x, e_i\...
- 6,255
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37
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Integral over a sphere in $R^n$
Let $\sigma$ be the Lebesgue measure on the unit sphere $\mathbb S^{n-1}$ of $\mathbb R^n$.
Let $\Sigma$ be a semi-definite positive symmetric matrix in dimension $n$.
Is it possible to get a closed-...
- 373
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61
views
Proof that the volume function is $\sigma$-additive
A $\textbf{box}$ $Q\subset \mathbb{R}^n$ is defined as $Q:= (a_1,b_1) \times \ldots \times (a_n,b_n)$ and the same with closed or half opened intervals where $a_i < b_i \in \mathbb{R}$ and in case ...
- 327
1
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90
views
Differentiability of an integral depending on a parameter
Let $f$ be a function in $L^1 ([0,1])$, and for $y \in [0,1]$ consider $$F(y) = \int_{[0,1]} (1+ |f|)^y dx$$
Is $F$ differentiable in $(0,1)$? If it is, what is its derivative?
I know that that, given ...
- 11
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95
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Example of Hilbertian norm on the space of radon measures
Assume we are given compact subset $X \subset \mathbb{R}^n$. Consider the space of radon measures $M(X,\mathbb{R})$. I'm trying to find some Hilbert norm on this space either globally or locally. I ...
- 53
1
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50
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Parameter dependent functions integral
I got this question on a measure theory exam today, and after hours of discussing with my colleagues, im still quite confused. I have been able to prove the first point, but I am having trouble with ...
- 23
2
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1
answer
41
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specific confusion on average integral
any idea why this is true? I am not able to figure out.
Given $f \in W^{1,p}(B(x, R))$, we want to prove that
$$
\left| \frac{1}{|B_{2^{-l}}(x)|} \int_{B_{2^{-l}}(x)} f - \frac{1}{|B_{2^{-l-1}}(x)|} \...
- 427