請教各位先進, 我這樣做可以嗎?
{fn} Cauchy in L^p p≧1
claim: {fn} uniformly Cauchy almost everywhere.
pf:
for arbitrary δ>0
since m{|fn-fm|≧1/j}≦ (j^p )||fn-fm||^p → 0
there exist N_k s.t. n,m≧N_k
m{|fn-fm|≧1/j}＜δ/2^k
moreover, we can ask N_k ↑
let Z_k={|fn-fm|≧1/j n,m≧N_k} so we have m(Z_k)＜δ/2^k
let Z=∪Z_k, then mZ≦δ, this implies mZ=0
Therefore, outside Z
{fn} uniformly Cauchy.
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