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78 Polynomial Approximation of Differential Equations
We only discuss the one dimensional case. Let I =]a,b[, a < b, be a bounded
interval. The Lebesgue measure µ associated to a subset J ⊂
Ī is a non negative
real number. However, not all the subsets are measurable, though these cases are quite
involved. Let us begin by considering very simple sets. Briefly, we review the most
significant cases.
The empty set J = ∅ has measure zero. If J ⊂
Ī contains a single point or a
finite number of points, then µ(J) := 0. When J is an interval with endpoints c and d,
a ≤ c < d ≤ b, it is natural to require that
µ(]c,d[) = µ([c,d[) = µ(]c,d]) = µ([c,d]) := d − c.
When J is the union of a finite number m of disjoint subintervals Ki, 1 ≤ i ≤ m, then
its measure is given by µ(J) := m
i=1 µ(Ki). Sets of this form are called elementary
sets. Union, intersection and difference of two elementary sets is still an elementary set.
The measure of an elementary set is easily reduced to the sum of lengths of one or more
segments.
A further generalization is to assume that J is the union of a countable collection
of disjoint intervals Ki ⊂ Ī, i ∈ N; we then set µ(J) = ∞
i=0 µ(Ki). This series
converges since it is bounded by the value b−a. We note that, if the intervals Ki, i ∈ N,
are not all disjoint (i.e., there exist i,j ∈ N such that Ki ∩Kj = ∅), then we can replace
the family {Ki}i∈N by the family {K ∗ i }i∈N such that J = ∪ ∞ i=0 K∗ i and the K∗ i ’s
are disjoint. In this case, we have the inequality
(5.1.1) µ(J) =
∞
i=0
µ(K ∗ i ) ≤
∞
i=0
µ(Ki).
Now, let J ⊂ Ī be a generic subset. Inspired by (5.1.1), we can always define the outer
measure µ ∗ of J. This is given by
(5.1.2) µ ∗
∞
∞
(J) := inf µ(Ki)
J ⊂
i=0
where the lower bound is taken over all the coverings of J by countable families of
intervals Ki, i ∈ N. For elementary sets the outer measure µ ∗ coincides with the actual
measure µ.
i=0
Ki
,