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Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\, \sum_{i=1}^n a_i = 1,\, x_i \in K \Big\},$$ define the $\sigma$-convex hull as $$\sigma\text{-}\mathrm{co}(K) = \Big\{ \sum_{i=1}^\infty a_i x_i : a_i \geq 0,\, \sum_{i=1}^\infty a_i = 1,\, x_i \in K \Big\},$$ where the summation is to be understood as convergence of the sequence in the topology of $X$.

I would like to understand conditions under which $\sigma\text{-}\mathrm{co}(K)$ is exactly the closure of $\mathrm{co}(K)$. In particular, does this property hold for any separable normed space $X$, or are further constraints on $X$ required?

The motivation for this question is Choquet's theorem, which allows one to write $$\overline{\mathrm{co}}(K) = \Big\{ \int x d\mu(x) : \mu \in M(K) \Big\}$$ with $M(K)$ standing for probability measures on $K$ for any compact subset $K$ in a normed space. I would like to understand the "countable" version of this theorem as presented above, but I could not find any references nor do I have an idea about how one could prove it.

Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\, \sum_{i=1}^n a_i = 1,\, x_i \in K \Big\},$$ define the $\sigma$-convex hull as $$\sigma\text{-}\mathrm{co}(K) = \Big\{ \sum_{i=1}^\infty a_i x_i : a_i \geq 0,\, \sum_{i=1}^\infty a_i = 1,\, x_i \in K \Big\},$$ where the summation is to be understood as convergence of the sequence in the topology of $X$.

I would like to understand conditions under which $\sigma\text{-}\mathrm{co}(K)$ is exactly the closure of $\mathrm{co}(K)$. In particular, does this property hold for any separable normed space $X$, or are further constraints on $X$ (and $K$?) required?

The motivation for this question is Choquet's theorem, which allows one to write $$\overline{\mathrm{co}}(K) = \Big\{ \int x d\mu(x) : \mu \in M(K) \Big\}$$ with $M(K)$ standing for probability measures on $K$ for any compact subset $K$ in a normed space. I would like to understand the "countable" version of this theorem as presented above, but I could not find any references nor do I have an idea about how one could prove it.

Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\, \sum_{i=1}^n a_i = 1,\, x_i \in K \Big\},$$ define the $\sigma$-convex hull as $$\sigma\text{-}\mathrm{co}(K) = \Big\{ \sum_{i=1}^\infty a_i x_i : a_i \geq 0,\, \sum_{i=1}^\infty a_i = 1,\, x_i \in K \Big\},$$ where the summation is to be understood as convergence of the sequence in the topology of $X$.

I would like to understand conditions under which $\sigma\text{-}\mathrm{co}(K)$ is exactly the closure of $\mathrm{co}(K)$. In particular, does this property hold for any separable normed space $X$? Also, is compactness of $K$ always necessary?

The motivation for this question is Choquet's theorem, which allows one to write $$\overline{\mathrm{co}}(K) = \Big\{ \int x d\mu(x) : \mu \in M(K) \Big\}$$ with $M(K)$ standing for probability measures on $K$ for any compact subset $K$ in a normed space. I would like to understand the "countable" version of this theorem as presented above, but I could not find any references nor do I have an idea about how one could prove it.

Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\, \sum_{i=1}^n a_i = 1,\, x_i \in K \Big\},$$ define the $\sigma$-convex hull as $$\sigma\text{-}\mathrm{co}(K) = \Big\{ \sum_{i=1}^\infty a_i x_i : a_i \geq 0,\, \sum_{i=1}^\infty a_i = 1,\, x_i \in K \Big\},$$ where the summation is to be understood as convergence of the sequence in the topology of $X$.

I would like to understand conditions under which $\sigma\text{-}\mathrm{co}(K)$ is exactly the closure of $\mathrm{co}(K)$. In particular, does this property hold for any separable normed space $X$, or are further constraints on $X$ required?

The motivation for this question is Choquet's theorem, which allows one to write $$\overline{\mathrm{co}}(K) = \Big\{ \int x d\mu(x) : \mu \in M(K) \Big\}$$ with $M(K)$ standing for probability measures on $K$ for any compact subset $K$ in a normed space. I would like to understand the "countable" version of this theorem as presented above, but I could not find any references nor do I have an idea about how one could prove it.

Let $K$ be a locally convex topological space, and let $X \subset K$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(X) = \{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\, \sum_{i=1}^n a_i = 1,\, x_i \in X \},$$ define the $\sigma$-convex hull as $$\sigma\text{-co}(X) = \{ \sum_{i=1}^\infty a_i x_i : a_i \geq 0,\, \sum_{i=1}^\infty a_i = 1,\, x_i \in X \},$$ where the summation is to be understood as convergence of the sequence in the topology of $K$.

I would like to understand conditions under which $\sigma\text{-co}(X)$ is exactly the closure of $\text{co}(X)$. In particular, does this property hold for any separable normed space $K$? Also, is compactness of $K$ always necessary?

The motivation for this question is Choquet's theorem, which allows one to write $$\overline{\text{co}}(X) = \{ \int x d\mu(x) : \mu \in M(X) \}$$ with $M(X)$ standing for probability measures on $X$ for any compact $X$ in a normed space. I would like to understand the "countable" version of this theorem as presented above, but I could not find any references nor do I have an idea about how one could prove it.

Let $X$ be a locally convex topological space, and let $K \subset X$ be a compact set. Recalling that the standard convex hull is defined as $$\text{co}(K) = \Big\{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\, \sum_{i=1}^n a_i = 1,\, x_i \in K \Big\},$$ define the $\sigma$-convex hull as $$\sigma\text{-}\mathrm{co}(K) = \Big\{ \sum_{i=1}^\infty a_i x_i : a_i \geq 0,\, \sum_{i=1}^\infty a_i = 1,\, x_i \in K \Big\},$$ where the summation is to be understood as convergence of the sequence in the topology of $X$.

I would like to understand conditions under which $\sigma\text{-}\mathrm{co}(K)$ is exactly the closure of $\mathrm{co}(K)$. In particular, does this property hold for any separable normed space $X$? Also, is compactness of $K$ always necessary?

The motivation for this question is Choquet's theorem, which allows one to write $$\overline{\mathrm{co}}(K) = \Big\{ \int x d\mu(x) : \mu \in M(K) \Big\}$$ with $M(K)$ standing for probability measures on $K$ for any compact subset $K$ in a normed space. I would like to understand the "countable" version of this theorem as presented above, but I could not find any references nor do I have an idea about how one could prove it.