Expectation of random variable $f(w) = 1-w$.
Is this right?
$Fx(x) = \mu_f$
$ \left\{\begin{matrix} \mu(\emptyset) = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for \ x < 0 \\ \mu([1-x,1)) = x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for \ \ 0 \leq x \leq 1\\ \mu([0,1]) = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for \ x > 1 \end{matrix}\right.$
So $$E(f) = {\int_\Omega f(w) \mu_f(dw) = \int_R f(w) \mu_f(dw)}$$ $$=\int_R (1-x) \mu_f(dx) $$ $$=\int_{[0,\infty)} (1-x) \mu_f(dx) $$ $$=1 \mu_f([0,\infty)) -x[0,\infty) = 1-x$$