= Solution
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 give
$$
\mathbb E\lambda(A_1\cap A_2)=\mathbb E\lambda(A_1)+\mathbb E\lambda(A_2)-\mathbb E\lambda(R)=m/2+m/2-m=0.
$$
Finally <Tonelli theorem> gives
$$
\boxed{\int_{\mathbb R^2}\mathbb P(y\in A_1\cap A_2)\,dy=\mathbb E\lambda(A_1\cap A_2)=0.}
$$
Finiteness of $m$ is what justifies the subtraction of <expectations>; an infinite-minus-infinite calculation would not establish this conclusion.
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