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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 $G$.2012-12-18

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