All Questions
34
questions
5
votes
0
answers
136
views
Property of vector-valued measure
Let $B$ be a Banach space, let $(X,\mathcal{A})$ be a measurable space, and let $\mu:\mathcal{A}\to B$ be a vector-valued measure of bounded variation.
In general, if $B$ doesn't have the Radon-...
- 8,127
0
votes
1
answer
57
views
In a Banach Space, when is $\sum |T(x_n)| < \infty$ for every sequence where $\sum x_i$ converges?
Given a Banach space $X$, and an operator $T \in X^*$, when does it hold that
$$\sum_{i=1}^\infty |T(x_n)| < \infty$$
for every sequence $\{x_n\}$ in $X$ such that $\sum_1^\infty x_n$ converges? As ...
- 1,307
0
votes
0
answers
102
views
Showing $\frac{1}{x \ln (x)}$ is in $L^p$ for all $p>0$
I was trying to find a function which is in $L^p[0,1]$ for all $p$ but not in $L^{\infty}[0,1]$. I asked my professor, when he said it's $\frac{1}{x \ln (x)}$. However, I am having a hard time to show ...
- 420
1
vote
1
answer
79
views
If $\int_A f \mathrm d \mu \in F$ for every measurable set $A$, then $f$ takes values on $F$ almost surely
Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $f:X \to E$ be $\mu$-integrable and $F$ a non-empty subset of $E$....
- 17.6k
6
votes
3
answers
298
views
Bochner integral: Is $f=g$ $\mu$-a.e. if their integrals are equal on every measurable set?
Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite complete measure space, and $(E, |\cdot|_E)$ a Banach space. Assume $f,g:X \to E$ are $\mu$-integrable such that
$$
\int_A f \mathrm d \mu = \int_A g \...
- 17.6k
1
vote
0
answers
186
views
A generalized version of dominated convergence theorem for Banach spaces
I'm trying to prove this generalized version of dominated convergence theorem for Banach space. Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space, $(E, |\cdot|)$ a Banach space, and $p \in ...
- 17.6k
1
vote
0
answers
47
views
Completing a measure space does not affect the integrability of a function
Let $(X, \mathcal A, \mu)$ be a measure space and $(E, | \cdot |)$ a Banach space.
$f \in E^{X}$ is called $\mu$-simple if $f = \sum_{k=1}^n e_k 1_{A_k}$ where $e_k \in E$ and $(A_k)_{k=1}^n$ is a ...
- 17.6k
2
votes
0
answers
100
views
A question on Bochner's integral.
Let $(S, \mathcal A, \mu)$ be a measure space and $x : S \longrightarrow X$ be a function to a Banach space $X.$ Then $x$ is Bochner integrable if there exists a sequence of simple measurable ...
- 2,912
1
vote
0
answers
45
views
Integration along a curve in Banach Spaces, any Measure Theory equivalent?
I'm studying Differential Calculus from the A. Avez book, and there's a topic about integration in Banach spaces, the thing is that I noticed that the definition given here is like the Riemann ...
- 63
1
vote
1
answer
275
views
L^1 is a Banach space with integral of the absolute value norm
I have to show that if $(X,\mathcal{M}, \mu)$ is a complete measure space (if $E\in \mathcal{M}$ has measure $0$, then every subset is measurable and has measure $0$, then $\mathcal{L}^1(\mu)$ with ...
- 157
3
votes
0
answers
97
views
Radon Nikodym property (Banach valued) is separably determined $\sigma-$finite case
Im reading Analysis in Banach spaces (https://www.springer.com/gp/book/9783319485195) and I was trying to provide a different demostration of lemma 1.3.17, which is used in order to prove that the ...
1
vote
1
answer
37
views
Weak compactness of nonnegative part of unit ball of $L^1$
Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and let $B$ be the unit ball of $L^1(\Omega,\mathfrak{A},\mu;\mathbb{R})$. Suppose that
$$
B_+=\big\{u\in B\;|\;u\geq 0\;\:\text{$\mu$-a.e.}\big\}
$$
...
- 1,009
1
vote
1
answer
252
views
The dual of $\mathcal{L}_{\mathbb{R}}^{1}$ is $\mathcal{L}_{\mathbb{R}}^{\infty}$?
Let $(E,\mathcal{A},\mu)$ be a finite measure space. I have a problem understanding the differences between spaces $\mathcal{L}_{\mathbb{R}}^{1}$ (the space of all classes functions $f\colon E\to\...
- 1,259
1
vote
1
answer
165
views
Is there a Radon-Nikodym derivative of a L^2-valued vector measure
Let $\Omega$ be a measurable space and $X$ a separable dual Banach space. $X$ is known to have the Radon-Nikodym property: If $\mu$ is a $\sigma$-additive measure with range in $X$ and $\nu$ a $\sigma$...
- 1,297
0
votes
1
answer
71
views
Is this construction of Bochner integral problematic?
I'm reading the construction of Bochner integral from this Wikipedia page.
The construction seems circular and thus problematic to me. We are calculating an integral containing $f$, i.e. $$\lim_{...
- 17.6k