↑Suppose now that μ and λ are both complete σ finite measures. Let
denote the completion of this measure. Let the larger measure space be
. Thus if E ∈σ
, it follows there exists a set A ∈ σ
E ∪ N = A where
. Now argue that for λ a.e.
y,x →XN is measurable because it is equal to zero
μ a.e. and μ is complete.
makes sense and equals zero. Use to argue that for λ a.e. y,x →XE is
measurable and equals ∫
dμ. Then by completeness of λ,y →∫
λ measurable and
Use this to give a generalization of the above Fubini theorem. Prove that if
f is measurable with respect to the σ algebra, σ
where the iterated integrals make sense.