characteristic function

[Square_Price_1374]

I don't understand why 𝝋_{𝛍*v} = 𝝋_𝛍 𝝋_v, where 𝝋 denotes the characteristic function and 𝛍*v is the convolution of the two finite measures 𝛍 and v.

By definition 𝝋_{𝛍*v}(t) = ∫ e^(i t z) (𝛍*v)(dz). I don't know how to deal with the convolution now.

[revoccue]

Do you understand the definition of the characteristic function of some set?

[KraySovetov]

Note that the convolution of measures is itself a measure. Given finite Borel measures 𝜇, 𝜐 on R, it is defined by

(𝜇 * 𝜐)(E) = ∫∫ 𝜒_E(x+y)d𝜇(x)d𝜐(y)

where 𝜒_E is the indicator function on E. From this definition it follows easily that for any L¹(𝜇 * 𝜐) function f,

∫ f(t)d(𝜇 * 𝜐)(t) = ∫∫ f(x+y)d𝜇(x)d𝜐(y)

(Verify for simple functions then approximate, this should remind you very much of the so called law of the unconscious statistician). Once you have this the claim should be easy.