27.7. CONVOLUTION AND SUMS 757

(µ ∗ν)(A)≡∫

Eν (A− x)dµ (x) .

Therefore, if s is a simple function, s(x) = ∑nk=1 ckXAk (x) ,∫

Esd (µ ∗ν) =

n

∑k=1

ck

∫E

ν (Ak− x)dµ (x) =∫

E

n

∑k=1

ckν (Ak− x)dµ (x)

=∫

E

n

∑k=1

ckXAk−x (y)dν (y)dµ (x) =∫

E

∫E

s(x+ y)dν (y)dµ (x)

Approximating with simple functions it follows that whenever f is bounded and measurableor nonnegative and measurable,∫

Ef d (µ ∗ν) =

∫E

∫E

f (x+ y)dν (y)dµ (x) (27.14)

Therefore, letting Z = X +Y, and λ the distribution of Z, it follows from independence ofX and Y that for t∗ ∈ E ′,

φ λ (t∗)≡ E

(eit∗(Z)

)= E

(eit∗(X+Y )

)= E

(eit∗(X)

)E(

eit∗(Y ))

But also, it follows from 27.14

φ (µ∗ν) (t∗) =

∫E

eit∗(z)d (µ ∗ν)(z) =∫

E

∫E

eit∗(x+y)dν (y)dµ (x)

=∫

E

∫E

eit∗(x)eit∗(y)dν (y)dµ (x)

=

(∫E

eit∗(y)dν (y))(∫

Eeit∗(x)dµ (x)

)= E

(eit∗(X)

)E(

eit∗(Y ))

Since φ λ (t∗) = φ (µ∗ν) (t

∗) , it follows λ = µ ∗ν .Note the last part of this argument shows the characteristic function of a convolution

equals the product of the characteristic functions. ■