# Density of $C(X,\operatorname{co}\{\delta_y\}_{y \in Y})$ in $C(X,\mathcal{P}(Y))$

Let $$X,Y$$ be locally-compact Polish spaces, equip the set $$\mathcal{P}(Y)$$ of probability measures on $$Y$$ with the weak$$^{\star}$$ topology (topology of convergence in distribution), and equip $$C(X,\mathcal{P}(Y))$$ with the compact-open topology. Let $$\operatorname{co}(\{\delta_y\}_{y \in Y})$$ denote the set of finite convex combinations of point-masses on $$Y$$.

(When) is the subset $$C(X,co(\{\delta_y\}_{y \in Y}))$$ dense in $$C(X,\mathcal{P}(Y))$$?


The answer to your question is yes.

Indeed, recall that the weak$$^{\star}$$ topology on $$\PP(Y)$$ is metrized by the Lévy--Prokhorov metric $$L$$ defined by the condition: for any real $$h>0$$ and any $$P,Q$$ in $$\PP(Y)$$, $$$$L(P,Q)\le h\iff\forall B\in\B(Y)\ P(B)\le Q(B_h)+h,$$$$ where $$\B(Y)$$ is the Borel $$\sigma$$-algebra over $$Y$$ and $$B_h$$ is the $$h$$-neighborhood of $$B$$.

Let $$\supp P$$ denote the support of $$P$$. Let $$\PP(K):=\{Q\in\PP(Y)\colon\supp Q\subseteq K\}$$.

Lemma 1: Take any compact $$\K\subseteq\PP(Y)$$ and any real $$\ep>0$$. Then there is a compact $$K\subseteq Y$$ such that for each $$P\in\K$$ there is some $$Q_P\in\PP(K)$$ such that $$$$\sup_{P\in\K}L(P,Q_P)\le\ep.$$$$

Proof: By Prokhorov's theorem, for each real $$\de\in(0,1)$$ there is a compact $$K\subseteq Y$$ such that for each $$P\in\K$$ we have $$P(K)\ge1-\de$$. For all $$B\in\B(Y)$$, let then $$$$Q_P(B):=\frac{P(B\cap K)}{P(K)}.$$$$ Then for all $$P\in\K$$ and all $$B\in\B(Y)$$ we have $$$$Q_P(B)\le\frac{P(B)}{1-\de}\le P(B)+\frac\de{1-\de}=P(B)+\ep$$$$ if $$\de=\ep/(1+\ep)$$. So, indeed $$L(P,Q_P)\le\ep$$ for all $$P\in\K$$. $$\Box$$

Let $$\PP_{fin}(Y)$$ denote the set of all $$Q\in\PP(Y)$$ with a finite support.

Lemma 2: Take any compact $$\K\subseteq\PP(Y)$$ and any real $$\ep>0$$. Then for each $$P\in\K$$ there is some $$R_P\in\PP_{fin}(Y)$$ such that $$$$\sup_{P\in\K}L(P,R_P)\le2\ep.$$$$

Proof: Let $$K\subseteq Y$$ and $$Q_P\in\PP(K)$$ for $$P\in\K$$ be as in Lemma 1. Since $$K$$ is compact, there is a finite Borel-measurable partition $$(C_{\ep;j})$$ of $$K$$ such that for each $$j$$ there is a point $$y_j\in Y$$ such that $$C_{\ep;j}$$ is contained in the ball $$B_\ep(y_j)$$ of radius $$\ep$$ centered at $$y_j$$.

For any $$P\in\PP(Y)$$ and any $$B\in\B(Y)$$, let $$$$Q_P^\ep(B):=\sum_j Q_P(C_{\ep;j})1_{y_j\in B}=Q_P(C_{\ep;B}),$$$$ where $$$$C_{\ep;B}:=\bigcup_{j\colon y_j\in B}C_{\ep;j}.$$$$ Then $$Q_P^\ep\in\PP_{fin}(Y)$$. Moreover, $$L(Q_P^\ep,Q_P)\le\ep$$ for all $$P\in\PP(Y)$$, because $$C_{\ep;B}\subseteq B_\ep$$ for all $$B\in\B(Y)$$. Letting now $$R_P:=Q_P^\ep$$, we see that Lemma 2 follows by Lemma 1. $$\Box$$

Finally, take any function $$P_\cdot\in C(X,\PP(Y))$$, which is a continuous map $$X\ni x\mapsto P_x\in\PP(Y)$$. Take also any compact $$K_X\subseteq X$$. Then the image $$\K:=\K_{K_X}:=P_\cdot(K_X)$$ of the compact $$K_X$$ under the continuous map $$P_\cdot$$ is compact. So, by Lemma 2, for each real $$\ep>0$$ and each $$x\in K_X$$ there is some $$R_{\ep;x}\in\PP_{fin}(Y)$$ such that $$$$\sup_{x\in K_X}L(P_x,R_{\ep;x})\le2\ep.$$$$ This means that indeed $$C(X,\PP_{fin}(Y))$$ is dense in $$C(X,\PP(Y))$$, in the required sense.

• @JohnZornSu : I don't think you can make $f_k$ and $g$ continuous. You may want to ask this question in a separate post. – Iosif Pinelis Sep 18 '20 at 12:19