To define Pn+1 from Pn, we let......
We define a <the> function f by <by setting> f=......
We define the set T to consist of those f for which......
We define a map to be simple if......
Now M is defined to be the set of all sums of the form......
Here F is only defined up to an additive constant.
Here u+ and u- are the positive and negative parts of u, as defined in Section 5.
As defined in Section 1, these are structures of the form......
The function f so defined satisfies......
The notion of backward complete is defined analogously by exchanging the roles of f and f-1.
The fact that the number T(p) is uniquely defined, even though p is not, enables us to define the nullity of A as follows.
The other two defining properties of a σ-algebra are verified in the same manner.