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80 Polynomial Approximation of Differential Equations
Two measurable functions f and g coincide almost everywhere (abbreviated by a.e.),
when the measure of the set {x ∈ Ī| f(x) = g(x)} is zero. In this situation f and g
are called equivalent.
We have now collected enough ingredients to define the Lebesgue integral of a
measurable function. Let us start with the integral of a simple function f. In this case,
f attains at most a countable set of values γi, i ∈ N, and the sets Ai := {x ∈ Ī| f(x) =
γi}, i ∈ N, are measurable. Then we have
(5.1.5)
I
f dx :=
provided the series is absolutely convergent.
∞
i=0
γi µ(Ai),
In general, let f be a measurable function; then there exists a sequence of simple func-
tions fn, n ∈ N, uniformly convergent to f. Thus we set
(5.1.6)
I
f dx := lim
n→∞
fn dx,
I
provided the limit exists. In this case, the limit does not depend on the approximating
sequence {fn}n∈N.
The sum of two integrable functions is an integrable function. Moreover, for continuous
or monotone functions, the Lebesgue and Riemann integrals coincide.
We have the implication
(5.1.7)
I
|f| dx = 0 ⇐⇒ f ≡ 0 a.e.
This shows that point values, achieved on sets whose measure is zero, are not meaningful
for the evaluation of the integral. This observation is crucial. In the following, besides
a given integrable function f, we shall not distinguish among functions equivalent to
f. This is because we shall work with norms of f based on the Lebesgue integral, and
these are not affected by modifications of f on a set of measure zero.