Let (X, A, μ) be a measure space and 1 ≤ p1 ≤ p2 ≤ ∞.
Show:
(a) If μ(X) < ∞, then Lp2(X) ⊆ Lp1(X)
and ||f||p1 ≤ μ(X)^(1/p1 - 1/p2) ||f||p2 for all f ∈ Lp2(X).
(b) If X = [0, 1], A = M(m), and μ = m is the Lebesgue measure,
then Lp1(X) ⊆ Lp2(X) does not hold.
(c) If X = R, A = M(m), and μ = m is the Lebesgue measure, then neither L1(X) ⊆ L2(X) nor L2(X) ⊆ L1(X) holds.