※ 引述《kemowu (小展)》之銘言：
: r
: 1. If (X,M,μ) is a measure space and if f in L (X,M,μ) for some 0＜r＜∞.
: p
: Show that lim ∫ |f| dμ = μ{x in X | f(x)≠0}.
: p→0 X
( ∫ = ∫_X )
Let A = { |f| > 1 }, B = { 0 < |f| ≦1}. Then ∫|f|^p=∫_A |f|^p + ∫_B |f|^p.
∫_B |f|^p tends to μ(B) by the monotone convergent theorem.
Since f is in L^r then ∞ > ∫_A |f|^r≧ μ(A) and |f| is finite a.e..
Consider |f|^p tends to 1 pointwise a.e. and μ(A) is finite,
we can obtain ∫_A |f|^p tends to μ(A) by using the dominated convergent
theorem. Hence, we can show that ∫_A |f|^p and ∫_B|f|^p converges to
μ(A) and μ(B), respectively.
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