Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 33 / 5 / b

Path continuity makes each time-segment range compact and therefore Borel measurable. The membership event is jointly measurable in the sample and spatial variable: distance to the range equals the infimum of $|B_t-y|$ over a countable dense set of times. We may consequently apply the Tonelli theorem to its indicator.

Put $m=\mathbb{E}\lambda (R)<\infty$, using the allowed integrability hypothesis. Since $R=A_1\cup A_2$, finite-measure inclusion-exclusion and the preceding expectation identities giveFinally Tonelli theorem givesFiniteness of $m$ is what justifies the subtraction of expectations; an infinite-minus-infinite calculation would not establish this conclusion.