Page 120 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 10: Measurable Functions

Every set on R.H.S. is measurable. Notes
Therefore E ( > ) is measurable.
A
Hence is measurable.
A

Note The above theorem asserts that the characteristic function of non-measurable sets
are non-measurable even though the domain set is measurable.

10.1.6 Simple Function

A real valued function is called simple if it is measurable and assumes only a finite number of
values.
If is simple and has the values , , … , then
1 2 n
n
= i A i
i 1
where A = {x : (x) = }
i i
and A A is a null set.
i j
Thus we can always express a simple function as a linear combination of characteristic function.

Notes

(i) is simple A s are measurable.
i
(ii) sum, product and difference of simple functions are simple.
(iii) the representation of as given above is not unique.
But if is simple and { , , ……, } is the set of non-zero values of f, then
1 2 n
n
= i A i
i 1
where A = {x : (x) = }
i i
This representation of is called the Canonical representation. Here A s are disjoint
i
and s are distinct and non-zero.
i
(iv) Simple function is always measurable.

10.1.7 Step Function

A real valued function S defined on an interval [a, b] is said to be a step function if these is a
partition a = x < x … < x = b such that the function assumes one and only one value in each
o 1 n
interval.

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