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Nov 4, 2015 at 14:38 comment added Nina Holden Thanks very much for the reply. It turned out the result of Bass and Burdzy was sufficient. We first prove continuity of $f\mapsto \int_0^T 1_{B_t\geq f(t)}dt$, where $f\in Lip(M)$ and we use the supremum norm on $Lip(M)$. Then we prove Lipschitz continuity of this map on a dense countable set of $Lip(M)$ by using the result of Bass and Burdzy.
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Nov 3, 2015 at 22:19 history answered user253775 CC BY-SA 3.0