112
II - Convergence: Discrete variables
::=: ... r::::.-- ____
'"
I{I
I
"
t-== --. . . . . . . . r\
[7 '"
I{I
'"
1/--- ......... .
--I{I
······1\ I , -----..... LJ · · · · ·
a
b
fig. 3.
An economical method of ensuring this is to impose the following hypothesis
on I: lor any r > 0, there exists a step function

(11.2)
one says then that I is a regulated function on the interval I. If one then
chooses 1/J(x) = graphs of

the various intervals Ik on which the function

pairwise disjoint, the sum of their lengths is the length b - a of I, so that
(11.3)
m(1/J) - m( if r < c/(b - a). The second condition of the axiom (IV bis) is thus satisfied.
The linearity of m(f) reduces to the two following propositions: (i) m(cf) =
em(f) for every constant c E 1R; (ii) m(f + g) = m(f) + m(g) for any regulated I and g. (i) is obvious for a step function and one proves the general
case thanks to (3). Additivity is shown similarly, observing that, if one has
two step functions each of which I and g, the general case following in the limit.
Example 1. Suppose that you wish to calculate the area L(a, b) lying between
the x-axis, the curve y = l/x = I(x) and the verticals with abscissae a > 0
and b > a. Let us choose an integer n, put q = (b/a)l/n, whence b = aqn, and
divide the interval [a, b] by the points a = aqo, aq, aq2, ... ,aqn = b. Consider
the step function

::; I since I is decreasing and
m( (aq - a)/aq + (aq2 - aq)/aq2 + ... + (aqn - aqn-l)/aqn =
=
n(q - l)/q = n(c 1 / n - 1)/c 1 / n where c = b/a.