Brownian occupation time

The occupation time of a measurable set $A$ through time $t$ is ${\int }_0^t{1}_{\{B_s\in A\}}ds$. The occupation-times formula expresses such integrals through local time of a semimartingale. Recurrence and the Strong Markov property imply that one-dimensional Brownian motion spends an unbounded total time in every nonempty open interval.

For a continuous semimartingale $X$ with local time of a semimartingale $L_t^a(X)$,for every nonnegative measurable function $f$.