a) A partition P of R is a family of rectangles (R ) J
J JfJ'
finite, satisfying the following:
0 0
i) for j,j'cJ, j j', R. (-h R = 0 (i.e., any two distinct
rectangles in (R.) are either disjoint or intersect only
J
on their boundaries)
ii) H = U R,.
JEJ J
(This is a straightforward extension of the notion of
partition of an interval [a,b]C R).
b) Let P = (R ) jc, Q = (R.) be two partitions of R. We say
j iJ l eT1
that Q is a refinement of P if, for each Rj E P, there exists
a family of rectangles from Q forming a partition of R..
J
(Note: It is evident from the definitions that for j j'.
The two families from Q forming partitions of R. and R., must
be disjoint.)
We show next that any two partitions of a given rectangle R have a
common refinement, as is the case in one dimension. The main step in
this, and a result we shall use again in its own right, is the
following:
2
Lemma 2.2.2. Let R [(s,t),(s',t')] be a rectangle in R2, and
P = (R )jEJ be a partition of R. Then there exist partitions
o: s = so < ss < ... < s = s' of [s,s'] and T: t = tO < tI <
... < t = t' of [t,t'] such that the family Q of rectangles of the
n
form [(s ,t ),(sp+1,t q)], O p < m, 0 5 q < n, is a refinement of P.
Remark. A partition of R constructed from partitions o,T of [s,s']
and [t,t'], respectively, as Q is above is called a grid on R. We