6.T. Show that if f∈L∞(X,X,µ), then |f(x)|≤∥f∥∞ for almost all x. Moreover,
if A <∥f∥∞then there exists a set E∈X with µ(E)>0 such that |f(x)|>A for all x∈E.
6.U. Show that if f∈Lp, 1≤p≤∞, and g∈L∞, then the product fg∈Lp and
∥fg∥p ≤ ∥f∥p∥g∥∞
6.T. and 6.U. (attached)
See attached file for full problem description.
keywords: integration, integrates, integrals, integrating, double, triple, multiple, integrable, integrability, measurable, measurability
A Lebesgue measure is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.