Page 139 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes where h is a bounded measurable function such that
m {x : h (x) 0} < 

Theorem 4: Let f be a non-negative measurable function. Show that f 0 implies f = 0 a.e.

Proof: Let be any measurable simple function such that

f.
Since f = 0 a.e. on E
0 a.e.

(x) dx 0
E
Taking supremum over all those measurable simple functions f, we get

f dx 0 … (1)
E
Similarly let be any measurable simple function such that f
Since f = 0 a.e.
0 a.e.

(x) dx 0
E
Taking infimum over all those measurable simple functions f, we get

f dx 0 … (2)
E
From (1) and (2), we get

f dx = 0.
E

Conversely, let f dx = 0
E

1
If E = x : f(x) , then
n n

1
f dx > (x) dx
n E n
E E
1 1
But (x) dx = mE (By definition)
n E n n n
E

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