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For $\mu\in \Delta(X)$ and $x\in X$ we have $x\in \operatorname{supp}(\mu)$ if and only if for any positive integer $n$ the function $\delta_{x,n}:=\big(1/n - d(\cdot,x)\big)_+$ has positive integral wrto the measure $\mu$. Therefore $$\{(x,\mu)\in X\times\Delta(X)\,:\, x\in \operatorname{supp}(\mu)\}=\bigcap_{n\in\mathbb{N}_+}\bigcup_{k\in\mathbb{N}_+} \{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\}\, . $$ For any $n$ and $k$ the set $\{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\} $ is closed, just because both the map $X\ni x\mapsto \delta_{x,n }\in \big(C_b(X),\|\cdot\|_\infty\big)$ and the pairing $C_b(X)\times\Delta(X)\ni(f,\mu) \mapsto \langle \mu,f \rangle$ are continuous ($\Delta(X)$ being endowed with the weak topology). This shows that the above set (the graph of the support map) is a Borel set; note that by consequence $\Theta'$ is Borel even for measurable $\mathbb{P}:\Theta\to\Delta(X)\,.$

For $\mu\in \Delta(X)$ and $x\in X$ (which I assume to be a metric space with distance $d$) we have $x\in \operatorname{supp}(\mu)$ if and only if for any positive integer $n$ the function $\delta_{x,n}:=\big(1/n - d(\cdot,x)\big)_+$ has positive integral wrto the measure $\mu$. Therefore $$\{(x,\mu)\in X\times\Delta(X)\,:\, x\in \operatorname{supp}(\mu)\}=\bigcap_{n\in\mathbb{N}_+}\bigcup_{k\in\mathbb{N}_+} \{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\}\, . $$ For any $n$ and $k$ the set $\{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\} $ is closed, just because both the map $X\ni x\mapsto \delta_{x,n }\in \big(C_b(X),\|\cdot\|_\infty\big)$ and the pairing $C_b(X)\times\Delta(X)\ni(f,\mu) \mapsto \langle \mu,f \rangle$ are continuous ($\Delta(X)$ being endowed with the weak topology). This shows that the above set (the graph of the support map) is a Borel set; note that by consequence $\Theta'$ is Borel even for measurable $\mathbb{P}:\Theta\to\Delta(X)\,.$

For $\mu\in \Delta(X)$ and $x\in X$ we have $x\in \operatorname{supp}(\mu)$ if and only if for any positive integer $n$ the function $\delta_{x,n}:=\big(1/n - d(\cdot,x)\big)_+$ has positive integral wrto the measure $\mu$. Therefore $$\{(x,\mu)\in X\times\Delta(X)\,:\, x\in \operatorname{supp}(\mu)\}=\bigcap_{n\in\mathbb{N}_+}\bigcup_{k\in\mathbb{N}_+} \{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\}\, . $$ For any $n$ and $k$ the set $\{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\} $ is closed, just because both the map $X\ni x\mapsto \delta_{x,n }\in \big(C_b(X),\|\cdot\|_\infty\big)$ and the pairing $X\times\Delta(X)\ni(x,\mu) \mapsto \langle \mu,\delta_ {x,n } \rangle$ are continuous.

This shows that the above set (the graph of the support map) is a Borel set; note that by consequence $\Theta'$ is Borel even for measurable $\mathbb{P}:\Theta\to\Delta(X)\,.$

For $\mu\in \Delta(X)$ and $x\in X$ we have $x\in \operatorname{supp}(\mu)$ if and only if for any positive integer $n$ the function $\delta_{x,n}:=\big(1/n - d(\cdot,x)\big)_+$ has positive integral wrto the measure $\mu$. Therefore $$\{(x,\mu)\in X\times\Delta(X)\,:\, x\in \operatorname{supp}(\mu)\}=\bigcap_{n\in\mathbb{N}_+}\bigcup_{k\in\mathbb{N}_+} \{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\}\, . $$ For any $n$ and $k$ the set $\{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\} $ is closed, just because both the map $X\ni x\mapsto \delta_{x,n }\in \big(C_b(X),\|\cdot\|_\infty\big)$ and the pairing $C_b(X)\times\Delta(X)\ni(f,\mu) \mapsto \langle \mu,f \rangle$ are continuous ($\Delta(X)$ being endowed with the weak topology). This shows that the above set (the graph of the support map) is a Borel set; note that by consequence $\Theta'$ is Borel even for measurable $\mathbb{P}:\Theta\to\Delta(X)\,.$

For $\mu\in \Theta$ and $x\in X$ we have $x\in \operatorname{supp}(\mu)$ if and only if for any positive integer $n$ the function $\delta_{x,n}:=\big(1/n - d(\cdot,x)\big)_+$ has positive integral wrto the measure $\mu$. Therefore $$\{(x,\mu)\in X\times\Delta(X)\,:\, x\in \operatorname{supp}(\mu)\}=\bigcap_{n\in\mathbb{N}_+}\bigcup_{k\in\mathbb{N}_+} \{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\}\, . $$ For any $n$ and $k$ the set $\{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\} $ is closed, just because both the map $X\ni x\mapsto \delta_{x,n }\in \big(C_b(X),\|\cdot\|_\infty\big)$ and the pairing $X\times\Delta(X)\ni(x,\mu) \mapsto \langle \mu,\delta_ {x,n } \rangle$ are continuous.

This shows that the above set (the graph of the support map) is a Borel set; note that by consequence $\Theta'$ is Borel even for measurable $\mathbb{P}:\Theta\to\Delta(X)\,.$

For $\mu\in \Delta(X)$ and $x\in X$ we have $x\in \operatorname{supp}(\mu)$ if and only if for any positive integer $n$ the function $\delta_{x,n}:=\big(1/n - d(\cdot,x)\big)_+$ has positive integral wrto the measure $\mu$. Therefore $$\{(x,\mu)\in X\times\Delta(X)\,:\, x\in \operatorname{supp}(\mu)\}=\bigcap_{n\in\mathbb{N}_+}\bigcup_{k\in\mathbb{N}_+} \{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\}\, . $$ For any $n$ and $k$ the set $\{(x,\mu)\,:\, \langle\mu,\delta_{x,n }\rangle\ge 1/k\} $ is closed, just because both the map $X\ni x\mapsto \delta_{x,n }\in \big(C_b(X),\|\cdot\|_\infty\big)$ and the pairing $X\times\Delta(X)\ni(x,\mu) \mapsto \langle \mu,\delta_ {x,n } \rangle$ are continuous.

This shows that the above set (the graph of the support map) is a Borel set; note that by consequence $\Theta'$ is Borel even for measurable $\mathbb{P}:\Theta\to\Delta(X)\,.$