Stochastic Programming - Index of

22 STOCHASTIC PROGRAMMING
{
• in IR 1 b − a if a ≤ b,
: µ(I [a,b) )=
0 otherwise,
{
• in IR 2 (b1 − a
: µ(I [a,b) )=
1 )(b 2 − a 2 ) if a ≤ b,
0 otherwise,
{
• in IR 3 (b1 − a
: µ(I [a,b) )=
1 )(b 2 − a 2 )(b 3 − a 3 ) if a ≤ b,
0 otherwise.
Analogously, in general for I [a,b) ⊂ IR k with arbitrary k, wehave
⎧
⎪⎨
µ(I [a,b) )=
⎪⎩
k∏
(b i − a i ) if a ≤ b
i=1
0 else.
(4.2)
Obviously for a set A that is the disjoint finite union of intervals, i.e.
A = ∪ M n=1 I(n) , I (n) being intervals such that I (n) ∩ I (m) = ∅ for n ≠ m,
we define its measure as µ(A) = ∑ M
n=1 µ(I(n) ).Inordertomeasureaset
A that is not just an interval or a finite union of disjoint intervals, we may
proceed as follows.
Any finite collection of pairwise-disjoint intervals contained in A forms
a packing C of A, C being the union of those intervals, with a welldefined
measure µ(C) as mentioned above. Analogously, any finite collection
of pairwise disjoint intervals, with their union containing A, formsacovering
D of A with a well-defined measure µ(D).
Take for example in IR 2 the set
A circ = {(x, y) | x 2 + y 2 ≤ 16, y≥ 0},
i.e. the half-circle illustrated in Figure 8, which also shows a first possible
packing C 1 and covering D 1 . Obviously we learned in high school that the
area of A circ is computed as µ(A circ )= 1 2 × π × (radius)2 =25.1327, whereas
we easily compute µ(C 1 )=13.8564 and µ(D 1 ) = 32. If we forgot all our
wisdom from high school, we would only be able to conclude that the measure
of the half-circle A circ is between 13.8564 and 32. To obtain a more precise
estimate, we can try to improve the packing and the covering in such a way
that the new packing C 2 exhausts more of the set A circ and the new covering
D 2 becomes a tighter outer approximation of A circ .ThisisshowninFigure9,
for which we get µ(C 2 )=19.9657 and µ(D 2 )=27.9658.
Hence the measure of A circ is between 19.9657 and 27.9658. If this is still
not precise enough, we may further improve the packing and covering. For
the half-cirle A circ , it is easily seen that we may determine its measure in this
way with any arbitrary accuracy.
In general, for any closed bounded set A ⊂ IR k , we may try a similar
procedure to measure A. DenotebyC A the set of all packings for A and by