Proof: As mentioned in Lemma 1, the topology of weak convergence on P is metrizable with the L´evy-Prohorov metric [15] and defined as for any µ, ν ∈ P , where A ( δ ) denotes the set of all x ∈$_{R}$ which lie a d -distance less than δ from A . Similarly as in the proof of Lemma 1, it suffices to show that Λ$_{0}$( γ, q$_{0}$ ) is both tight and closed in P . The tightness can be shown by the same arguments as in Lemma 1. In the following, we prove that Λ$_{0}$( γ, q$_{0}$ ) is closed in P . Let B$_{m}$ = $^{[}$− 1 $_{m}$, 1 m ] for m = 1 , 2 , . . . . Let { µ$_{n}$ } ∞ n =1 be a convergent sequence in Λ$_{0}$( γ, q$_{0}$ ) with limit µ$_{0}$ . For any m ∈$_{N}$ , there exists an n$_{m}$ such that L ( µ$_{n}$, µ$_{0}$ ) < 1 m for all n > n$_{m}$ . By the definition of L in (125), we have for any m ∈$_{N}$ and n > n$_{m}$ , and For any n ∈$_{N}$ $^{⋃}${ 0 } , we have so for any m ∈$_{N}$ , there exists an n ′ m such that µ$_{n}$ ( B$_{m}$ ) ≤ µ$_{n}$ ( { 0 } ) + 1 $_{m}$. Therefore, according to (126) and (127), for all m ∈$_{N}$ and n > max { n$_{m}$, n $^{′}$$_{m}$} , Thus we have µ$_{0}$ ( { 0 } ) = q$_{0}$ by letting m → ∞ . Moreover, let f ( x ) = x 2 which is continuous and bounded below. By weak convergence [15, Sec- tion 3 . 1 ], we have 25 E$_{µ}$$_{0}$ $^{{}$X 2 } = ∫ f d µ$_{0}$ ≤ lim inf n →∞ ∫ f d µ$_{n}$ ≤ γ. (130) q$_{0}$ − 2 m ≤ µ$_{0}$ ( { 0 } ) ≤ q$_{0}$ + 2 m . (129) µ$_{n}$ ( { 0 } ) = µ$_{n}$$^{(}$ ∞ ⋂ m =1 B$_{m}$$^{)}$ = lim m $_{→∞}$µ$_{n}$ ( B$_{m}$ ) , (128) µ$_{n}$ ( { 0 } ) ≤ µ$_{0}$ ( B$_{m}$ ) + 1 m . (127) µ$_{0}$ ( { 0 } ) ≤ µ$_{n}$ ( B$_{m}$ ) + 1 m , (126) L ( µ, ν ) = inf $^{{}$δ : µ ( A ) ≤ ν ( A ( δ $^{)}$) + δ and ν ( A ) ≤ µ ( A ( δ $^{)}$) + δ for all A ⊆ B } (125)