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Results tagged with measure-theory
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user 99469
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
1
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0
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190
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Is the domain space in Lusin's theorem required to be Hausdorff?
I'm reading a general version of Lusin's theorem, i.e.,
If $\mu$ is a finite Radon measure on $X$, and $Y$ is a second countable topological spaces, then for any Borel-measurable function $f:X\to Y$ …
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6
votes
1
answer
197
views
Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-alge...
Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Then we have Theorem 1.40 in Rudin's Real and Complex Analysis, i.e., …
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1
vote
1
answer
265
views
Let $f$ be convex and $A$ a Borel subset of $\mathbb R^d$ on which $f$ is differentiable. Is...
Let $X := \mathbb R^d$, $\lambda^d$ be the $d$-dimensional Lebesgue measure on $X$, and $f:X \to \mathbb R$ convex. Then there is a Borel set $N \subset X$ such that $\lambda^d (N) = 0$ and $f$ is dif …
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0
votes
1
answer
76
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Decay rate of $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$
We fix $p \in [1, \infty)$. We have for every $f \in L^p (\mathbb R^d)$ that $\lim_{|z| \to \infty} \|1_{B(z, 1)} f\|_{L^p} =0$. I wonder if there is an estimate of above decay, i.e.,
Is there a meas …
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1
vote
2
answers
126
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Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integr...
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3
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Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integr...
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2
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1
answer
61
views
Approximate a non-negative function which is measurable in product $\sigma$-algebra
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0
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1
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107
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If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all ...
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral.
Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, $1 \ …
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0
votes
1
answer
277
views
Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl.
THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach space. …
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1
vote
Accepted
Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
Below is my formalization of @Nik's hints to finish the proof.
Let's prove that
$$
\sum_{m=1}^M \|H_m\|^q_{L_{p}(\mu_m, X)^*} \le \|H\|^q_{L_{p}(\mu, X)^*} \quad \forall M \in \mathbb N^*.
$$
Let $\O …
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2
votes
0
answers
92
views
A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged
I have recently read about about disintegration theorem, i.e.,
Disintegration theorem Let
$X$ be a Polish space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X …
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1
vote
1
answer
61
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Let $c: X \times Y \to \overline{\mathbb R}$ be $\gamma$-measurable. Is $c_x:Y \to \overline...
Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$.
$f:X \to \overline{\mathbb R}$ is called $\mu$-si …
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0
votes
2
answers
102
views
How to construct this sequence that converges a.e. in product measure and that has a very pa...
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the …
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1
vote
How to construct this sequence that converges a.e. in product measure and that has a very pa...
First, we prove that for all $f \in S(X\times Y)$ and $\varepsilon >0$. There is $f_\varepsilon \in S(X\times Y)$ such that $g$ satisfies $(*)$ and that $\lambda (A) \le \varepsilon$ where $A := \{f …
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1
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1
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272
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Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
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