Page 129 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
where M = sup f(x) ,
i
x i 1 x x i
m = inf f(x)
i
x i 1 x x i
We then define the upper Riemann integral of f by

b
R f(x) dx inf S
a
where the infimum is taken over all possible sub-divisions of [a, b].
Similarly, we define the lower Riemann integral

b
R f(x) dx sup S
a
The upper integral is always at least as large as the lower integral, and if the two are equal, we
say that f is Riemann integrable and we call this common value the Riemann integral of f.
It will be denoted by

b
R f(x)dx
a

Note By a step function we mean function s.t.
(x) = x [x , x ]
i i–1 i
for some sub-division of [a, b] and some set of constant then
i
b x 1 x n
(x) dx = (x)dx (x)dx
a x o x n 1

x 1 x 2 x n
= dx dx dx
1 2 n
x o x 1 x n 1

= (x – x ) + (x – x ) + … + (x – x )
1 1 0 2 2 1 n n–1 n
n
= i (x i x ) … (1)
i 1
i 1
with this in mind, we see that

b
R f (x) dx = inf p (f)
a

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