User joe king - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:17:14Z http://mathoverflow.net/feeds/user/22531 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92706/closed-balls-in-banach-spaces Closed balls in Banach spaces joe king 2012-03-30T19:27:26Z 2012-03-30T19:27:26Z <p>Let $(X,\|\cdot\|)$ be a Banach space and let $B(x; r)$ be the <em>closed</em> ball with radius $r$ around $x\in X$. Let $H(X)$ be the collection of all closed and bounded subsets of $X$.</p> <blockquote> <p><strong>I want to show the following:</strong> For any finite Borel measure $\mu$, the map $X\to\mathbf{R}\colon x\mapsto \mu\big(B(x,r)\big)$ is Borel measurable.</p> </blockquote> <p><em>This is what I've done so far:</em></p> <p>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:</p> <p>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 $\mathrm{dist}\big(y,B(x,r)\big)=\inf\limits_{z\in X}\|y-z\|$</p> <p><em>I've established the following:</em></p> <ol> <li>For $x\in X$ and $r>0$ the sequence $f_x^n(y)$ converges pointwise to the indicator function $\mathbf{1}_{B(x,r)}(y)$.</li> <li>$f_{x_m}^n\to f_x^n$ for any sequence $\lbrace x_m\colon m\in\mathbf{N}\rbrace$ with limit $x\in X$.</li> <li>The mapping $g\colon X\to H(X)\colon x\mapsto B(x,r)$ is continuous (w.r.t. the the Hausdorff distance).</li> </ol> <p>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.</p> <p>Can someone help me out? Is there some reference (book, article, etc.) were I can find this?</p>