For the second, let rn be the countable rational numbers in the interval [0,1] and define the functions.

gn(x)= {1,0 if x= r1,r2,r2,... rn and g(x) = 1 if x = irrational and g(x)=0 if x=rational

then g(n) converges point wise to g

the limit function is not integrable

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