... real numbers with α > 1 and1α+1β=1. For x ∈ Lα(X, Σ,p) and y ∈ Lβ(X, Σ,p) with x α> 0and yβ> 0, use the Arithmetic-Geometric Mean Inequality (Exercise 64) with a1=|x(ω)|αxαand ... α)=(S).Exercise 28.H(a)(Non-atomicity of ) Show that, for any A ∈ B(R) with (A) > 0,there exists a B ∈ B(R) with B ⊆ A and 0 < (B) < (A).(b) Give an example of a measure on B(R) ... beautiful survey on the origins of the theory oftheLebesguemeasureandintegral.)The situation may at first seem somewhat reminiscent of Cantor’s countability theory viewingcountably infinite sets...