Page 158 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 12: General Convergence Theorems

Notes
But lim f = lim f lim f
n n n
E E E

Hence f = lim f .
n
E E

Corollary: Let {u } be a sequence of integrable functions on E such that u converges a.e. on
n n
n 1
n
E. Let g be a function which is integrable on E and satisfy u g a.e. on E for each n. Then
i
i 1

u is integrable on E and u n u .
n
n
n 1 E n 1 n 1 E
n
Proof: Let u f .
i n
i 1
Applying Lebesgue Dominated Convergence Theorem for the sequence {f }, we get
n
u u
n n
E n 1 n 1 E
Corollary: If f is integrable over E and {E } is a sequence of disjoint measurable sets such that
i
E E , then
i
i 1

f f
E i 1 E i
Proof: Since {E } is a sequence of disjoint measurable sets, we may write.
i
f = f
E i
i 1

The function f. E i is integrable over E since f E i |f| and |f| is integrable over E. Moreover

n
f |f|, n N
E i
i 1
Thus the conditions of above corollary are satisfied and hence

f = f
E i
E E i 1

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