To define $P_{n+1}$ from $P_n$, we let ......
We define a $\langle$the$\rangle$ function $f$ by setting $f=Tg$. ......
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 the 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 $\sigma$-algebra are verified in the same manner.
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