I'm reading Alan Weir's "Lebesgue Integration and Measure".
In exercise 8 on page 30 he talks about "...an increasing sequence of step functions $\{\phi_n\}$..." and "...an increasing sequence of sets $\{S_n\}$..." However, he doesn't seem to define what these are.
Can anyone help?
On
The first one means a pointwise increasing sequence of step functions. He is probably approximating a function from below by a pointwise increasing sequence of step functions so that the limit is almost everywhere the desired function.
An increasing sequence of sets means a nested sequence of sets: every set contains the previous one.
An increasing sequence of functions is a sequence $\{\phi_n\}$ such that $\phi_{n+1}(x)\geq\phi_n(x)$ for all $x$ and $n$. An increasing sequence of sets $\{S_n\}$ has the property that $S_n\subseteq S_{n+1}$ for all $n$.