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Would be grateful if someone could provide a hint or an appropriate reference for the following.

Notation:

$\mathcal{P}(\mathbb{R}^n)$- Space of probability measures on $\mathbb{R}^n$

$g_n\stackrel{\Gamma}{\longrightarrow}g$ denotes gamma convergence of appropriate space

Question

Let $(\rho_n)$ be a sequence in $\mathcal{P}(\mathbb{R}^n)$ and $f_n:\mathbb{R}^n\rightarrow\mathbb{R}$ a sequence of functions. Define, $F_n(\rho):=\int\limits_{\mathbb{R}^n}f_n(x)\rho(dx)$.

Prove that: $f_n\stackrel{\Gamma}{\longrightarrow}f$ in $\mathbb{R}^n \Longleftrightarrow F_n\stackrel{\Gamma}{\longrightarrow}F$ in $\mathcal{P}(\mathbb{R}^n)$.

Remarks: (1) I am looking to use narrow convergence of probability measures ($\rho_n$) in context of proving $\Gamma$ convergence of $F_n$.
(2) This usually pops up in existence of solution to some Fokker-Planck type of equation. More specifically it pops in Sandier-Serfaty framework for Gradient flows.

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Do you mean the Gamma convergence like in the link: en.wikipedia.org/wiki/%CE%93-convergence –  Beni Bogosel Apr 15 '13 at 21:59
    
Yeah that's the one. Apologies for not providing the definition in the question itself. –  UPS Apr 16 '13 at 7:31

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