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Let $G$ be an open set on some measure space $(\Omega,\Sigma, \mu)$. Show that the indicator function of $G$ is an increasing limit of a sequence of continuous functions.

I understand that this is a "simple text book exercise". I am only seeking for hints.

My idea (at least in 2D) is that we should construct some smooth trapezoidal-like functions, where the gradients near the endpoints of $G$ gets steeper and steeper.

Am I on the right track? Also, how do I write this out mathematically?

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In order to measure the distance to the edge, you could use open balls in the space, since it is measurable. Something like $f(x)=0$ outside and $f(x)=\min(1,\sup\{nr:B(x,r)\subseteq G\})$ on the inside of $G$. –  Mario Carneiro Dec 18 '12 at 2:46
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1 Answer 1

Every open set in a separable metric space is a union of a sequence of open balls. So start by working out how to write the characteristic function of an open ball as an increasing sequence of continuous functions. As you suggested, this can be done with trapezoidal functions that get steeper in a certain way. The endpoints of the trapezoids need to move to make the sequence increasing...

In the most general sense, a measure space consists of a set $\Omega$, a $\sigma$-algebra $\Sigma$, and a measure $\mu$. Because no topology is mentioned in this definition, it does not make sense to try to prove anything about "open" sets or "continuous" functions in a general measure space. You have to be talking about some class of topological measure spaces for those terms to make sense. In this way, the question is underspecified, because it does not specify what type of topological measure spaces are being considered. In general, whether every open set of a topological space is an increasing limit of continuous functions is a purely topological question, having little to do with measure theory.

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