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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?

Alok Singh
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  • Maybe something that that can help explain this is that for instance the rational numbers are dense in $\mathbb{R}$, so you still only need countable number of numbers to get as close to each value in the unit-interval as you want?(even though the unit interval is uncountable) – user119615 Dec 12 '15 at 01:15
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    Probably a good idea for you to get a little more convinced is to think of an example of a sequence of simple functions that converges to $ y = x $ on the unit interval. Moreover, think of the following: any real number is a limit of a sequence of rational numbers - of which there are only countably many – Jytug Dec 12 '15 at 01:17

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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!

grand_chat
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