Closed balls in Banach spaces

2012-03-30T19:27:26Z

Let $(X,\|\cdot\|)$ be a Banach space and let $B(x; r)$ be the closed ball with radius $r$ around $x\in X$. Let $H(X)$ be the collection of all closed and bounded subsets of $X$.

I want to show the following: For any finite Borel measure $\mu$, the map $X\to\mathbf{R}\colon x\mapsto \mu\big(B(x,r)\big)$ is Borel measurable.

This is what I've done so far:

I thought it would be good to relate the measure to the integral over the indicator function on the closed ball, i.e. write $$\mu\big(B(x,r)\big)=\int_X\mathbf{1}_{B(x,r)}(y)\ d\mu(y).$$ To do this, I need an appropriate sequence to approximate the indicator function:

Consider for $x,y\in X$ the sequence $\lbrace f_{x}^n\colon n\in\mathbf{N}\rbrace$ where $$f_x^n(y)=\max\Big(0, \Big[1-n\cdot\mathrm{dist}\big(y,B(x,r)\big)\Big]\Big),$$ where $\mathrm{dist}\big(y,B(x,r)\big)=\inf\limits_{z\in X}\|y-z\|$

I've established the following:

For $x\in X$ and $r>0$ the sequence $f_x^n(y)$ converges pointwise to the indicator function $\mathbf{1}_{B(x,r)}(y)$.
$f_{x_m}^n\to f_x^n$ for any sequence $\lbrace x_m\colon m\in\mathbf{N}\rbrace$ with limit $x\in X$.
The mapping $g\colon X\to H(X)\colon x\mapsto B(x,r)$ is continuous (w.r.t. the the Hausdorff distance).

How can I use the 3 properties to finish my proof? I know that part 1 gives that $$\mu\big(B(x,r)\big)=\int_X\mathbf{1}_{B(x,r)}(y)\ d\mu(y)=\int_X\lim_{n\to\infty}f_x^n(y)\ d\mu(y)$$ and part 3 implies that $g$ is measurable, since continuous maps are by definition Borel. I know that poinwise limits of Borel measurable functions are Borel measurable, so $\mathbf{1}_{B(x,r)}$ is Borel measurable.

Can someone help me out? Is there some reference (book, article, etc.) were I can find this?

User joe king - MathOverflow

Closed balls in Banach spaces