Page 121 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes

Notes
(i) Step function also assumes finite number of values like simple functions but the sets
{x : S (x) = C } are intervals for each i.
i
(ii) Every step function is also a simple function but the converse is not true.

1, x is rational
e.g. f : R R such that f (x) =
0, x is irrational
is a simple function but not step as the sets of rational and irrational are not intervals.

Theorem 6: If f and g are two simple functions then f + g is also a simple function.
Proof: Since f and g are simple functions and we know that every simple function can be
expressed as the linear combination of characteristic function.
f and g can be expressed as the linear combination of characteristic function.

m
f = i A i
i 1
m
and g = j B j
j 1
where A s and B s are disjoint.
i j
A = {x : f (x) = }
i i
B = {x : g (x) = }
j j
The set E obtained by taking all intersections A B from a finite disjoint collection of measurable
k i j
sets and we may write
n
f = a k E k
k 1
n
and g = b k E k
k 1
where n = mm .

n n
f + g = a b
k E k k E k
k 1 k 1
n
= ( a k b k ) E k
k 1
which is a linear combination of characteristic functions, therefore it is simple.

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