Can someone help me understand the following?

Question

I was reading George Lowthers notes on Stochastic Calculus and , he says the following but I cannot figure out what it exactly means?

http://almostsure.wordpress.com/2009/11/08/filtrations-and-adapted-processes/#comment-4727

He writes the following after defining a Previsible Process and a Predictable $\sigma$-algebra

Given any set of real-valued functions on a set, which is closed under multiplication, the set of functions measurable with respect to the generated sigma-algebra can be identitified as follows. They form the smallest set of real-valued functions containing the generating set and which is closed under taking linear combinations and increasing limits. So, for example, the predictable processes form the smallest set containing the adapted left-continuous processes which is closed under linear combinations and such that the limit of an increasing sequence of predictable processes is predictable.

Answer 1

It is an identification. Call the generated $\sigma$-algebra $A$, the set of functions being used for the generation $F_0$, and the set of $A$-measurable real-valued functions $F$. Then $A$ is the smallest $\sigma$-algebra such that:

1. Any function in $F_0$ is $A$-measurable.
2. Any linear combination of members of $F$ is $A$-measurable.
3. Any increasing limit of a sequence of members of $F$ is $A$-measurable.

The last sentence is an application of this identification.