Page 118 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 10: Measurable Functions

10.1.3 Equivalent Functions Notes

Definition: Two functions f and g defined on the same domain E are said to be equivalent on E,
written as f ~ g on E, if f = g a.e. on E, i.e. f (x) = g (x) for all x E – E , where E E with m (E )
1 1 1
= 0.
Theorem 2: If f, g : E R (E M) such that g (E).
Proof: Let be any real number and let E = E (f > ) and E = E (g > )
1 2
Then E E = (E – E ) (E – E )
1 2 1 2 2 1
= {x E : f (x) g (x)}
so that by given hypothesis we have
m (E E ) = 0.
1 2
This together with the fact that E is measurable
1
E is measurable.
2
Hence g (E).

10.1.4 Non-negative Functions

+
–1
Definition: Let f be a function, then its positive part, written f and its negative part, written f , are
defined to be the non-negative functions given by
+
–1
f = max (f, 0) and f = max (–f, 0) respectively.

Note f = f – f –1
+
and |f| = f + f –1
+
Theorem 3: A function is measurable iff its positive and negative parts are measurable.
Proof: For every extended real valued function f, we may write

1
+
f = [f + |f|]
2
1
and f –1 = [|f| – f]
2
But f is measurable then |f| is measurable and hence positive and negative parts of f i.e. f and
+
–
f are measurable.
+
–1
Conversely, let f and f be measurable.
Since f = f – f –1
+
Since we know that if f and g are measurable functions defined on a measurable set E then f – g
is measurable on E.
+
–1
Here f – f is measurable.
and hence f is measurable.

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