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Let $(\Omega,\mathfrak{F})$ be a measurable space and let $X$ be the set of all probability measures on $(\Omega,\mathfrak{F})$. Then fix a probability measure $\nu$; i there a topology that we can put on $(\Omega,\mathfrak{F})$ making the set $$ Y\triangleq \{ \mu \sim \nu \}, $$ closed?

Where two measures are said to be equivalent if and only if $\mu \ll \nu\ll \mu$; that is both measures are absolutely continuous with respect to each other.

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    What you can do is consider the topology determined by taking the complements of equivalence classes to be an open sub-base. This is (from definition) the the topology with the "least amount of open sets" so that these sets are closed. – 2017-01-23
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    Ya I thought of this, but this may not be a "nice topology" and then I would have to determine all its properties from scratch (might be a side-track from a research point of view) – 2017-01-23

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