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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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    I am working on this problem and I am stuck at (a), could u please give me some hint? I cannot figure out what it means "add in each of the intervals that are omitted in the first step of the construction which is another Cantor type set". Thanks in advance.2016-12-06

2 Answers 2

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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 $n\in\mathbb{N}$.