Let P be the set of probability measures μ on R such that R[X]⊂L1(μ). Let us consider the equivalent relation ∼ on P given by μ1∼μ2 if and only if for every P∈R[X],
∫Pdμ1=∫Pdμ2
(i.e. μ1 and μ2 share the same sequence of moments). We say that μ∈P is characterized by its moments when its equivalent class is a singleton. Every compactly supported probability measure on R belongs to P and is indeed characterized by its moments thanks to the Weierstrass density theorem. Beyond compactly supported probability measures, one may use the Carleman criterion. The following classical lemma is useful when using the moments method, for instance when proving the Wigner semicircle theorem.
Lemma 1 (From moments convergence to weak convergence) If μ,μ1,μ2,… belong to P with
limn→∞∫Pdμn=∫Pdμ
for every P∈R[X] and if μ is characterized by its moments then for every f∈Cb(R,R)
limn→∞∫fdμn=∫fdμ.
Proof: The convergence assumption implies that for every polynomial P
CP:=supn≥1∫Pdμn<∞.
Thus, by Markov's inequality, for every real R>0,
μn([−R,R]c)≤CX2R2
and therefore (μn)n≥1 is tight. As a consequence, by Prokhorov's theorem, it suffices now to show that if (μnk)k≥1 converges weakly to ν then ν=μ. Recall that the weak convergence here is also known as the narrow convergence and corresponds to the convergence for continuous bounded functions.
Let us show that μ=ν. Let us fix some P∈R[X] and a real number R>0. Let φR:R→[0,1] be continuous with 1[−R,R]≤φR≤1[−R−1,R+1]. We start from the decomposition
∫Pdμnk=∫φRPdμnk+∫(1−φR)Pdμnk.
Since (μnk)k≥1 tends weakly to ν we have
limk→∞∫φRPdμnk=∫φRPdν.
Additionally, by Cauchy-Schwarz's and Markov's inequalities,
|∫(1−φR)Pdμnk|2≤μnk([−R,R]c)∫P2dμnk≤CX2CP2R2.
On the other hand, we know that
limk→∞∫Pdμnk=∫Pdμ.
Therefore, we obtain
limR→∞∫φRPdν=∫Pdμ.
Using this for P2 we obtain by monotone convergence that P∈L2(ν)⊂L1(ν) and then by dominated convergence that
∫Pdν=∫Pdμ.
Since P is arbitrary and μ is characterized by its moments, it follows that μ=ν.