What does it mean for a sequence of measures $\mu_t$ to be weakly continuous in $t$?

Question

We have a measurable space equipped with a collection of (probability) measures $\mu_t$, $t\in \mathbb R^+$. What does it mean for $\mu_t$ to be weakly continuous in $t$ ? Is it something like

$$F(t)=\int f(x) d\mu_t(x)$$

is continuous in $t$ for all bounded continuous function $f$ ?

What do you mean by "continuous function $f$" in this context? — Ruy, Sep 05 '20 at 02:29
I don't really know, I was trying to make a definition based on what "weakly" means in other context. I guess if the measurable space has a topology, then it makes sense. Would that be the definition ? — W. Volante, Sep 05 '20 at 03:12
This definition would work if the topological space were compact, Hausdorff. — Stephen Montgomery-Smith, Sep 05 '20 at 04:13

score 1 · Accepted Answer · answered Sep 05 '20 at 23:15

Let $(X,\mathscr{B})$ be a topological space equipped with the Borel $\sigma$-algebra, $\mathcal{C}_b(X)$ be the collection of bounded continuous functions, and $\mathcal{C}_{00}(X)$ the collection of continuous functions of compact support.

Typically, if $X$ is a metric space, then weak continuity of the collection of probability measures $\{\mu_t:t\in\mathbb{R}_+\}$ means that $t\mapsto\mu_t$ is continuous as a map from $\mathbb{R}_+$ with the usual topology on the real line to $\mathcal{M}(X)$ (the space of finite measures) with the weak topology $\sigma(\mathcal{M}(X),\mathcal{C}_b(X))$. This is equivalent to what is suggested in the OP, that is, for any $t_0$,

$$\lim_{t\rightarrow t_0}\int_Xf\,d\mu_t=\int_Xf\,d\mu_{t_0}\,\qquad\forall f\in\mathcal{C}_b(X)$$

If $(X,\tau)$ is a (general) locally compact Hausdorff topological space, it is typical to say that the collection of probability measures $\{\mu_t:t\geq0\}$ is continuous if $t\mapsto\mu_t$ is continuous as a map from $\mathbb{R}_+$ to $\mathcal{M}(X)$ the latter equipped with the weak topology $\sigma(\mathcal{M}(X),\mathcal{C}_{00}(X))$, that is, for any $t_0$, $$\lim_{t\rightarrow t_0}\int_Xf\,d\mu_t=\int_Xf\,d\mu_{t_0}\,\qquad\forall f\in\mathcal{C}_{00}(X)$$
For other ore general topological spaces, or measurable spaces without a topological structure, other notions of weak continuity (weak topology on $\mathcal{M}(X)$) may be used, depending on applications.

Ruy · Answer 2 · 2020-09-05T14:26:30.137

If $X$ is a compact Hausdorff space then the dual of $C(X)$ is isomorphic to the space of all regular Borel measures with finite total variation. This is known as the Riesz representation Theorem.

Viewing measures as linear functionals, according to Riesz, allows one to transfer the weak topology over to the space of measures. This leads to the notion of weak topology for measures which is correctly described in the original post.

What does it mean for a sequence of measures $\mu_t$ to be weakly continuous in $t$?

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