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This is on the construction of a measurable function $f$ on $[0,1]$ such that every function $g$ that differs from $f$ only ona set of measure zero is discontinuous at every point.

The exercise is #37 from pp. 45 and it asks the following:

(a) Construct a measurable set $E \subset [0,1]$ such that for any non-empty opensub-interval $I$ in $[0,1]$, both sets $E \cap I$ and $E^{c} \cap I$ have positive measure.

(b) Show that $f = \chi_{E}$ has the property that whenever $g(x) = f(x)$ almost everywhere, then $g$ must be discontinuous at every point in $[0,1]$.

While I think I got (a) using the hint to consider Cantor-like sets, I am stuck at (b); thanks in advance for any help.

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Just checking, but you know that if answers are helpful you can both accept them and upvote them, right? :) – cardinal Mar 7 '12 at 0:15
up vote 1 down vote accepted

Suppose $g(x_0)=0$ (the other cases are similar). Let $\epsilon=\frac 1 2$, for all $\delta>0, \ E\cap (-\delta+x_0,x_0+\delta)$ has positive measure and since $f=g$ a.e. there is an $a\in E\cap (-\delta+x_0,x_0+\delta)\cap \{x:f(x)=g(x)\}$. Thus, $|a-x_0|<\delta$ and $|g(x_0)-g(a)|=1\geq \frac 1 2$

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HINT: Let $A=\{x\in[0,1]:g(x)=f(x)\}$. Fix $x\in[0,1]$. Show that there must be sequences $\langle x_n:n\in\mathbb{N}\rangle$ and $\langle y_n:n\in\mathbb{N}\rangle$ in $[0,1]$, both converging to $x$, such that $x_n\in E\cap A$ and $y_n\in Z\setminus A$ for each $n\in\mathbb{N}$.

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