# Thread: Showing a function is measurable

1. ## Showing a function is measurable

Let $X=(S,d)$ a banach space. I want to show the following: For any borel measure $\mu$, the map $\psi: X\to \mathbb{R}$ given by $x\mapsto \mu(\overline{B}_x(r))$ is measurable.

I believe this can be done in a number of ways, one of which is proving that the set $U_c:=\left\{x:\psi(x)\leq c\right\}$ for any $c$ is contained in $\mathcal{B}(X)$(borel sigma-algebra of $X$)

I think proving this, can not be done without using the fact that $d$ is induced by some norm $\left\|\cdot \right\|$. If we take for example $X= (\mathbb{R}, d_E)$ with Euclidian metric, ( $\mu$ lebesgue measure) it is obvious $U_c$ is contained in $\mathcal{B}(X)$ as $U_c = X$ or $U_c = \emptyset$ for any $c$. But for any random metric $d$, this is not obvious...

A norm-induced metric has (as I recall) 2 extra properties. 1. translation invariance $d(x+a,y+a)=d(x,y).$ and 2. $d(\alpha x,\alpha y) = |\alpha|d(x,y)$

I hope what I think is correct here...so suppose $\psi(x)= \alpha$ for some $\alpha >0$ then for any $y\in X$ we have $\psi(y)=\alpha$ as well...

It's just a hunch, i have no clue actually...as I don't know what properties $\mu$ might have. (except for the standard properties)

Any idea what I'm missing here?

Thank you

(good to see u guys back online though...btw: can it be implemented that $x^2$ is valid latex-code here, instead of wraps..it's sometimes
quite bothersome to write tags everywhere :P)

2. ## Re: Showing a function is measurable

Can yo use Fubini's theorem?