All Questions
Tagged with measure-theory integration
2,732
questions
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21
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Measurability of a family of parametric integrals assuming measurability of the integrand w.r.t. the parameter
Let $D$ and $E$ be measurable subsets of $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and $v : (x,t) \in D \times E \mapsto v(x,t) \in \mathbb{C}$.
Assume that the maps $v(\cdot, t)$, $t \in E$ ...
5
votes
1
answer
2k
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How to understand uniform integrability?
From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
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28
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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 ...
1
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68
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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 ...
4
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3
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2k
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Why does the triangle inequality for space $L^p$ with $0<p<1$ not hold?
It is said that triangle inequality for the space $L^p(\mathbb{R})$ space doesn't hold if $0<p<1$.
Does anyone know an example for this?
Also, what we can say, for example, about the quantity ...
1
vote
1
answer
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 ...
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 ...
0
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0
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32
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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 ...
0
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1
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39
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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:...
0
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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 $H$:
$$P_n x = \sum_{i=1}^n \langle x, e_i\...
0
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0
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64
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Find the limit $\lim_{k \rightarrow \infty } \int_0^k x^{-k} e^{x^2/k^2}\sin(x/k)~\mathrm dx$
Compute the limit$$\lim_{k \rightarrow \infty } \int_0^k x^{-k} e^{x^2/k^2}\sin(x/k)~\mathrm dx.$$
Completely stuck with this one. Some convergence theorem is obviously needed, but can't figure out ...
0
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39
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Integral over a sphere in $R^n$
Let $\sigma$ be the Lebesgue measure on the unit sphere $\mathbb S^{n-1}$ of $\mathbb R^n$.
Let $\Sigma$ be a semi-definite positive symmetric matrix in dimension $n$.
Is it possible to get a closed-...
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62
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Proof that the volume function is $\sigma$-additive
A $\textbf{box}$ $Q\subset \mathbb{R}^n$ is defined as $Q:= (a_1,b_1) \times \ldots \times (a_n,b_n)$ and the same with closed or half opened intervals where $a_i < b_i \in \mathbb{R}$ and in case ...
66
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4
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8k
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Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
1
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0
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91
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Differentiability of an integral depending on a parameter
Let $f$ be a function in $L^1 ([0,1])$, and for $y \in [0,1]$ consider $$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 ...