Me and some classmates are stuck on a problem from our real analysis textbook (Royden/Fitzpatrick 4th Ed.), which we've given some thought. The question is:
"Suppose $f$ is a real valued function on $\Bbb R%$ such that ${f^-}^1(c)$ is measurable for each $c$. Is $f$ necessarily measurable?"
Our thoughts are that it is not true. The only example of nonmeasurable set we know is the Vitali Set $\Bbb V$, and the only nonmeasurable function we know of is the characteristic function of $\Bbb V$, denoted $X_{\Bbb V}$.
$X_{\Bbb V}(x) = \cases{ 1 & \text{if } x\in \Bbb V\cr 0 & \text{if } x\notin \Bbb V } $
I was thinking that possibly ${X_{\Bbb V}^-}^1(c)$ is measurable for all $c\in\Bbb R$, but I wasn't sure. The range of ${X_{\Bbb V}}(x)$ is just $[0] \bigcup [1]$.... Is this suitable?
Perhaps a more interesting question...on the nature of nonmeasurable sets and functions. Are there alternate constructions of nonmeasurable sets and functions?