Page 117 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes (f + g) , (f – g) are measurable functions over E.
2
2
(f + g) – (f – g) is a measurable function over E.
2
2
1 2 2
(f g) (f g) is a measurable function over E.
4
fg is a measurable function over E.
(h) To prove f/g is measurable.
Let g vanish nowhere on E, so that

g (x) 0 x E.
1
exists.
g

1
Now we shall show that is measurable.
g
We have

E(g 0) if 0
1 1
E [E(g 0)] E g if 0
g
1
[E(g 0)] E(g 0) E g

1
Also finite union and intersection of measurable sets are measurable. Hence E is
g
measurable in every case.

1
Since f and are measurable.
g

1
(f) is measurable over E.
g
f
is measurable over E.
g

10.1.2 Almost Everywhere (a.e.)

Definition: A property is said to hold almost everywhere (a.e.) if the set of points where it fails to
hold is a set of measure zero.

Example: Let f be a function defined on R by

0, if x is irrational
f (x) =
1, if x is rational
Then f (x) = 0 a.e.

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