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Functional Spaces 79
We are now ready to give the definition of a Lebesgue measurable set. The subset J ⊂ Ī
is measurable if and only if
(5.1.3) µ ∗ (J) = µ( Ī) − µ∗ ( Ī − J).
Then we set µ(J) := µ ∗ (J). The idea is the following. The term on the left-hand
side of (5.1.3) is obtained by approximating J from outside with families of intervals.
The term on the right-hand side represents in some sense an approximation from inside.
When both the estimates coincide, then J is measurable.
It is possible to show that the union and intersection of a finite number of measurable
sets are still measurable. Moreover, and this is the main improvement with respect to
Riemann measure theory, union and intersection of a countable family of measurable
sets are still measurable. If the sets Ji, i ∈ N, are measurable and disjoint, then
(5.1.4) µ
∞
i=0
Ji
=
∞
i=0
µ(Ji).
For instance, the set of rational numbers J = Q ∩Ī, is not Riemann measurable. Being
the union of a countable collection of points, J is Lebesgue measurable and by (5.1.4)
µ(J) = 0. If I is an unbounded interval, then J ⊂
measurable for any bounded interval K ⊂
also take the value +∞.
Ī is measurable when J ∩ K is
Ī. In this situation we assume that µ can
We can now introduce the concept of measurable function. A function f : Ī → R
is measurable when the set {x ∈
Ī| f(x) < γ} is measurable for any γ ∈ R. It is
possible to prove that the sum and the product of two measurable functions are still
measurable.
A measurable function that assumes a finite or countable set of values in
Ī, is called
simple function. Any bounded measurable function f can be represented as a uniform
limit of simple functions fn, n ∈ N (this means that limn→∞ sup x∈I |fn(x)−f(x)| = 0).