12.9. PRODUCT MEASURES 317

Corollary 12.9.13 If f ∈ L1 (X×Y ) , then∫f d (µ×ν) =

∫ ∫f (x,y)dµdν =

∫ ∫f (x,y)dνdµ.

If µ and ν are σ finite, then if f is µ×ν measurable having complex values and either∫ ∫| f |dµdν < ∞ or

∫ ∫| f |dνdµ < ∞, then

∫| f |d (µ×ν)< ∞ so f ∈ L1 (X×Y ) .

Proof: Without loss of generality, it can be assumed that f has real values. Then

f =| f |+ f − (| f |− f )

2

and both f+ ≡ | f |+ f2 and f− ≡ | f |− f

2 are nonnegative and are less than | f |. Therefore,∫gd (µ×ν)< ∞ for g = f+ and g = f− so the above theorem applies and∫

f d (µ×ν) ≡∫

f+d (µ×ν)−∫

f−d (µ×ν)

=∫ ∫

f+dµdν−∫ ∫

f−dµdν

=∫ ∫

f dµdν .

It remains to verify the last claim. Suppose s is a simple function,

s(x,y)≡m

∑i=1

ciXEi ≤ | f |(x,y)

where the ci are the nonzero values of s. Then

sXRn ≤ | f |XRn

where Rn ≡ Xn×Yn where Xn ↑ X and Yn ↑ Y with µ (Xn) < ∞ and ν (Yn) < ∞. It follows,since the nonzero values of sXRn are achieved on sets of finite measure,∫

sXRnd (µ×ν) =∫ ∫

sXRndµdν .

Letting n→ ∞ and applying the monotone convergence theorem, this yields∫sd (µ×ν) =

∫ ∫sdµdν . (12.9.48)

Now let sn ↑ | f | where sn is a nonnegative simple function. From 12.9.48,∫snd (µ×ν) =

∫ ∫sndµdν .

Letting n→ ∞ and using the monotone convergence theorem, yields∫| f |d (µ×ν) =

∫ ∫| f |dµdν < ∞