= Atomless maxima on separated Brownian intervals
{title2=$\mathbb P(\sup_{[a,b]}B=\sup_{[c,d]}B)=0,\quad a<b<c<d$}
For <Brownian motion>, write the later maximum as $B_b+Z+H$, where $Z=B_c-B_b$ has a nondegenerate <normal distribution> and $H=\sup_{0\leq u\leq d-c}(B_{c+u}-B_c)$. The <independent increments> make $Z$ independent of the earlier history and of $H$. Conditional on those two objects, equality of the two maxima prescribes one value of $Z$, an event of probability zero. This argument uses the positive gap $c-b$ and does not assume independence of the two absolute maxima.
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