7.20 If f and g are real-valued functions such that f 2, g2 are measurable with respect to the σ-finite measure µ, prove the Schwarz inequality



fg dµ2

≤



f 2 dµ

g2 dµ.

[Hint: Write fg dµ = EQ(f/g), where Q is the probability measure with dQ =

g2dµ/ g2dµ, and apply Jensen’s inequality with ϕ(x) = x2.]