Page 114 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 10: Measurable Functions

10.1 Measurable Functions Notes

10.1.1 Lebesgue Measurable Function/Measurable Function

Definition: Let E be a measurable set and R* be a set of extended real numbers. A function
f : E R* is said to be a Lebesgue measurable function on E or a measurable function on E iff the
set
–1
E (f > ) = {x E : f (x) > } = f { , )} is a measurable subset of E R.

Notes
1. The definition states that f is a measurable function if for every real number , the
inverse image of ( , ) under f is a measurable set.
2. The measure of the set E (f > ) may be finite or infinite.
3. A function whose values are in the set of extended real numbers is called an extended
real valued function.
4. If E = R, then the set E (f > ) becomes an open set.

Example: A constant function with measurable domain is measurable.
Sol: Let f be a constant function defined over a measurable set E so that f (x) = x E.
Then for any real number ,

E, if c
E (f > ) =
, if c

The sets E and are measurable and hence E (f > ) is measurable i.e. the function f is measurable.
Theorem 1: Let f and g be measurable real valued functions on E, and c is a constant. Then each of
the following functions is measurable on E.

(a) f c (b) c f
(c) f + g (d) f – g
(e) |f| (f) f 2
(g) fg (h) f/g (g vanishes no where on E)
Proof: Let be an arbitrary real number.

(a) Since f is measurable and
E (f ± c > ) = E (f >  c),
the function f ± c is measurable.

(b) To prove c f is measurable over E.
If c = 0, then cf is constant and hence measurable because a constant function is measurable.

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