Find the standard representation of the function f defined by f(x)=[x] for −1≤x≤3, f(x)=0 otherwise. determine the integral R of fdu
I came across this question while studying and began to attempt it but my text had no other examples quite like, nor did I find any help elsewhere online. What steps need to be taken to show the standard representation or is that found when finding the Lebesgue integral of this function? any clarity will greatly help, thanks.
Actually, $f$ is a simple function. To be concrete, its standard representation is $$f=-\chi_{[-1,0)}+\chi_{[1,2)}+2\,\chi_{[2,3)}+3\,\chi_{[3,3]}.$$ And hence its Lebesgue integral is \begin{align*} \int_{\mathbb{R}} f\,\mathrm{d}\mu&=-1\cdot \mu([-1,0))+1\cdot \mu([1,2))+2\cdot \mu([2,3))+3\cdot \mu([3,3])\\ &=-1\times 1+1\times 1+2\times 1+3\times 0\\ &=2. \end{align*}