2020/07/27 by di Dio, Philipp J.
#44A60 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2007.13347
In measure theory several results are known how measure spaces are transformed into each other. But since moment functionals are represented by a measure we investigate in this study the effects and implications of these measure transformations to moment funcationals. We gain characterizations of moments functionals. Among other things we show that for a compact and path connected set K⊂ℝn there exists a measurable function g:K→ [0,1] such that any linear functional L:ℝ[x1,…,xn]→ℝ is a K-moment functional if and only if it has a continuous extension to some L:ℝ[x1,…,xn]+ℝ[g]→ℝ such that L:ℝ[t]→ℝ defined by L(td) := L(gd) for all d∈ℕ0 is a [0,1]-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function f:[0,1]→ K independent on L such that the representing measure μ of L provides the representing measure μ∘ f-1 of L. We also show that every moment functional L:V→ℝ is represented by λ∘ f-1 for some measurable function f:[0,1]→ℝn where λ is the Lebesgue on [0,1].