# When a family of measures provide continuity?

Consider a mapping $m: \mathcal{B}(X) \times P \rightarrow [0,1]$, where $X \subseteq \mathbb{R}^n$, $P \subseteq \mathbb{R}^m$, and $\mathcal{B}(X)$ denotes the Borel sets.

$\forall p \in P$, $m(\cdot, p)$ is a probability measure on $X$, i.e. $m(X,p) = 1$ for all $p \in P$.

Consider $f: X \rightarrow [0,1]$ measurable.

Define $$F(p) := \int_{X} f(x) m(dx,p)$$

Under which - "weak" - conditions on $m$, $F$ is continuous?

-
In what topology? –  Michael Greinecker Apr 20 '13 at 19:55
Euclidean. I guess. –  Adam Apr 20 '13 at 19:59
What do you mean by locally bounded? $f$ is actually bounded. When $p\mapsto m(S,p)$ is continuous for each $S\subset X$ measurable, it works. –  Davide Giraudo Apr 20 '13 at 20:02