# Continuity of the preimage of a set under a measure preserving continuous function #

In this file we prove that the preimage of a null measurable set `s : Set Y`

under a measure preserving continuous function `f : C(X, Y)`

is continuous in `f`

in the sense that `μ ((f a ⁻¹' s) ∆ (g ⁻¹' s))`

tends to zero as `f a`

tends to `g`

.

As a corollary, we show that
for a continuous family of continuous maps `f z : C(X, Y)`

,
a null measurable set `s`

, and a null measurable set `t`

of finite measure,
the set of parameters `z`

such that `f z ⁻¹' t`

is a.e. equal to `s`

is a closed set.

Let `X`

and `Y`

be R₁ topological spaces
with Borel σ-algebras and measures `μ`

and `ν`

, respectively.
Suppose that `μ`

is inner regular for finite measure sets with respect to compact sets
and `ν`

is a locally finite measure.
Let `f : α → C(X, Y)`

be a family of continuous maps
that converges to a continuous map `g : C(X, Y)`

in the compact-open topology along a filter `l`

.
Suppose that `g`

is a measure preserving map
and `f a`

is a measure preserving map eventually along `l`

.
Then for any finite measure measurable set `s`

,
the preimages `f a ⁻¹' s`

tend to the preimage `g ⁻¹' s`

in measure.
More precisely, the measure of the symmetric difference of these two sets tends to zero.

Let `f : Z → C(X, Y)`

be a continuous (in the compact open topology) family
of continuous measure preserving maps.
Let `t : Set Y`

be a null measurable set of finite measure.
Then for any `s`

, the set of parameters `z`

such that the preimage of `t`

under `f_z`

is a.e. equal to `s`

is a closed set.

In particular, if `X = Y`

and `s = t`

,
then we see that the a.e. stabilizer of a set is a closed set.