# Tagged Questions

60 views

### Measure Theory and Functional analysis exercise book

I'm looking for a big collection of exercises of functional analysis and measure theory. I know a lot of theory books which present some excercises (Brezis, Rudin, Lang, Royden, and others) but I was ...
36 views

33 views

### A Borel measure defines semi-continuous function?

Let $X$ be a metric space with outer measure $\mu$, which is assumed to be a Borel measure, i.e., all Borel sets are measurable. For a fixed subset $A\subset X$ (not necessarily measurable, but you ...
34 views

### spectral measure of non-empty and open set is non-zero proof

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf Part d) on page 322 and his proof appears on page 324. I didn't quite understand his proof so I had a go at ...
36 views

### invariant subspace partition

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf on page 327 Rudin says that M and M' are invariant subspaces. I'm guessing he means non-trivial so how does ...
30 views

### Bochner measurability

I have the following problem. Let $(\Sigma, \Omega, \mu)$ be a measure space and let $X$ be a Banach space. Take a function $f \colon \Omega \rightarrow \mathbf{B}(X)$, which takes values in space of ...
15 views

### Why Steiner Symmetrization makes a measurable set to a measurable one?

I find the Steiner Symmetrization is very useful in proving that the Hausdorff measure coincide with Lebesgue in the Euclidean space. However, I never saw anybody mention that the Steiner ...
239 views

### Does $f(x)\in L^1$ imply that $f'(x) \in L^1$?

Let $f(x)$ be defined for all real numbers differentiable function of one variable.We know that: $$\int_{-\infty }^{+\infty } |f(x)| \, dx\neq +\infty$$ Problem is to resolve if it is possible or not ...
27 views

### Norm of multiplication operator

I have that $(X,\Omega,\mu)$ is a sigma finite space, and I have that $g$ is a measurable function. Assume that $fg\in L^p$ for all $1\leq p\leq \infty$. I want to show that $g\in L^\infty$. My idea ...
21 views

I need to show that: If $f$ is continuous at $x_0$ iff $f^*(x_0)=f_*(x_0)$ where: $f^*(x_0)=\lim_{x \to x_0} \sup f(y)=\inf_{\epsilon > 0} \sup_{|y-x_0|<\epsilon}f(y)$ and $f_*(x_0)=\lim_{x ... 1answer 46 views ### A dense subalgebra of$C(X)$that separates points Any idea how to do this problem: If$X$is a compact Hausdorff space and$A$a subalgebra of$C(X)$, where$C(X)$is the algebra of all continuous functions, such that$A$contains the constant ... 0answers 49 views ### The support of Gaussian measure in Hilbert Space$L^2(S^1)$with covariance$(1-\Delta)^{-1}$Let$\mu$be Gaussian measure defined on Hilbert space$\mathcal{H}=L^2(S^1)$($S^1$- circle) by formula $$\int e^{(f,g)} d\mu(f) = e^{-\tfrac{1}{2}(g,C g) }.$$ The covariance operator$C$is ... 1answer 65 views ### Are$L_p$spaces of functions with separable support separable? Let$X$be a separable space. Is$L_p$$(X, \mu, V) a separable space? Here, (V, |\cdot|_V) is a normed space. And a norm of L_p(X, \mu, V) is:$$ \|f\|=\left(\int_X \big(\vert f\rvert_V\big)^p ...
I am looking for a Lemma that gives an equivalent formulation for a family of functions to be equi-integrable: is it true that if $\{f_j\}_j\in L^1$, then we can write $f_j=f^1_j+f_j^2\in L^1+L^p$, ...