Page 149 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes (i) f (x) f (x), x E – E ,
n 0
and (ii) m (E ) = 0.
0
12.1.2 Pointwise Convergence

Let <f > be a sequence of measurable functions on a measurable set E. Then <f > is said to
n n
converge “pointwise” in E, if a measurable function f on E such that
f (x) f (x) x E or
n
lt f (x) = f (x)
n n

12.1.3 Uniform Convergence, Almost Everywhere (a.e.)

Let <f > be a sequence of measurable functions defined over a measurable set E. Then the
n
sequence <f > is said to converge uniformly a.e. to f, if a set E E s.t.
n 0
(i) m (E ) = 0 and
0
(ii) <f > converges uniformly to f on the set E – E .
n 0
12.1.4 Bounded Convergence Theorem

Theorem 1: State and Prove: Bounded Convergence Theorem
Statement: Let {f } be a sequence of measurable functions defined on a set E of finite measure, and
n
suppose that there is a real number M such that |f (x)| M n and x. If f (x) = lim f (x) for each
n n n
x in E, then

f = lim f
n n
E E

Proof: Since f (x) = lim f (x) and f is measurable on E
n n n
E
f is also measurable on E
Let > 0 be given

Then measurable set A E with mA < and a positive integer N such that
4M

|f (x) – f (x)| < n N and x E – A
n
2mE

f f = (f f)
n n
E E E

(f f)
n
E

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