Page 385 - DMTH401_REAL ANALYSIS

Unit 32: The General Lebesgue Integral and Convergence in Measure

Substituting (2) and (3) in (1), we get Notes
-
E ò
E ò g - E ò f E ò g lim f n
Hence

E ò
E ò f lim f n ...(4)
Similarly by considering g + f , we get
n
E ò f lim E ò f n ...(5)

From (4) and (5), we get
lim f n E ò f lim f n ...(6)
E ò
E ò
But it is always true that
lim f n lim f n ...(7)
E ò
E ò
From (6) and (7)

E ò f = lim f n .
E ò
Hence the theorem.

Notes If we replace g by g ’s, we get the following generalization of the Lebesgue
n
Convergence theorem.

Theorem 3: Let {g } be a sequence of integrable functions which converges a.e to an integrable
n
function g. Let {f } be a sequence of measurable functions such that |f | g and {f } converges to
n n n n
f a.e.
If g = lim g ò n ,
ò

then f ò = lim f ò n .

32.3 Convergence in Measure

Definition: A sequence {f } of measurable functions is said to converge to f in measure if, given
n
 > 0, there is an N such that for all n N we have
m{x/|f(x) – f (x)|} < .
n
Remark: From this definition and littlewood’s third principle, it is clear that,
If {f } is a sequence of measurable functions defined on a measurable set E of finite measure and
n
f > f a.e, then {f } converges to f in measure.
n n

Example: Construct the sequence {f } as follows:
n


Let n = k + 2 , 0 k < 2 , and
–
–
Set f (x) = 1 if x [k2 , (k + 1) 2 ]
n

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