I've been thinking back to the proof that in $\mathbb{R}$, a measurable function $f:\mathbb{R}\to\mathbb{R}$ is the pointwise limit of increasing simple functions $s_n$.
As far as the intuitive picture of it goes, Pugh and Folland both have excellent visualizations that convince me that "morally", such a theorem is correct (and are also a great deal of help is making sense of the messy algebra).
However, when I think of a function such as the identity on the unit interval ("y=x" from grade school), it is definitely measurable as it's continuous, but the fact that it takes on uncountably many values in the interval jars with my understanding that a simple function can only take on finitely many (and in the limit, increases to a countable number of values taken on).
What part(s) am I really failing to grasp here?
Think of the decimal expansion of a number in the unit interval. Let $f_n(x)$ be the $n$-digit approximation to $x$. Clearly $f_n(x)$ converges pointwise to $f(x):=x$. But each $f_n$ takes only finitely many values, so each $f_n$ is simple!