Stochastic Programming - Index of

28 STOCHASTIC PROGRAMMING
Figure 11
Integrating a simple function.
Next let us briefly review integrals. Consider first IR k with A, its measurable
sets, and the natural measure µ, and choose some bounded measurable set
B ∈A. Further, let {A 1 , ···,A r } be a partition of B into measurable sets,
i.e. A i ∈A, A i ∩ A j = ∅ for i ≠ j, and ⋃ r
i=1 A i = B. Giventheindicator
functions χ Ai : B −→ IR defined by
χ Ai (x) =
{
1 if x ∈ Ai ,
0 otherwise,
we may introduce a so-called simple function ϕ : B −→ IR given with some
constants c i by
ϕ(x) =
r∑
c i χ Ai (x)
i=1
= c i for x ∈ A i .
Then the integral ∫ ϕ(x)dµ is defined as
B
∫
B
ϕ(x)dµ =
r∑
c i µ(A i ). (4.6)
In Figure 11 the integral would result by accumulating the shaded areas with
their respective signs as indicated.
i=1