Orthonormal bases for product measures

Let B be the Borel σ-algebra of R, and let B be the Borel σ-algebra of [−∞,∞] = R ∪ {−∞,∞}: the elements of B are those subsets of R of the form B,B ∪ {−∞}, B ∪ {∞}, B ∪ {−∞,∞}, with B ∈ B. Let (X,A , μ) be a measure space. It is a fact that if fn is a sequence of A → B measurable functions then supn fn and infn fn are A → B measurable, and thus if fn is a sequence of A → B measurable functions that converge pointwise to a function f : X → R, then f is A → B measurable. If f1, . . . , fn are A → B measurable, then so are f1 ∨ · · · ∨ fn and f1 ∧ · · · ∧ fn, and a function f : X → R is A → B measurable if and only if both f = f ∨ 0 and f− = −(f ∧ 0) are A → B measurable. In particular, if f is A → B measurable then so is |f | = f + f−. A simple function is a function f : X → R that is A → B measurable and whose range is finite. Let E = E(A ) be the collection of nonnegative simple functions. It is straightforward to prove that