5.1 Monotone Class Theorems
There are several versions of monotone class theorems, both for
families of sets and for functions. The two stated here are the
variants we shall use later. The statements are taken from
Dellacherie and Meyer [4].
Theorem 1.5.1. Let C be a ring of subsets of Q. Then the monotone
class M generated oy C is equal to the o-ring generated by C. If C is
an algebra, then M = o(C).
Theorem 1.5.2. Let H be a vector space of bounded real-valued
functions defined on Q, which contains the constants, is closed under
uniform convergence, and has the following property:
for every uniformly bounded increasing sequence (f ) of
positive functions from H, the function f = lim fn belongs
n
to H.
Let C be a subset of H which is closed under multiplication. The
space H then contains all bounded functions measurable with respect to
the a-field o(C).
The most frequent application of this theorem comes when we wish
to prove that a certain property holds for all bounded F-measurable
functions; it allows us to reduce to the case where f is the indicator
of a set. We shall see numerous examples of this theorem at work.
1.5.2 Liftings
Let (T,E,p) be a measure space, p0. Material for this section
comes from Dinculeanu [6, pp. 199-216]. We shall need these
properties in Chapter IV.