In the third edition of Rudin's *Real and Complex Analysis, Rudin states Lusin's Theorem in an unusual way, and I think there may be an error. Here, $X$ is a locally compact Hausdorff space and $\mu$ is a measure on $X$. Here is his statement:
Suppose $f$ is a complex integrable function on $X$, $\mu(A)<\infty$, $f(x)=0$ if $x\notin A$, and that $\epsilon>0$. Then there exists a $g\in C_{c}(X)$ such that $ \mu(\{x:f(x)\neq g(x)\})<\epsilon. $
There is a bit more about some additional conditions we may place on $g$, but the part I am concerned about is in the above statement. To see my problem, consider the case of when $X=\mathbb{R}^{1}$ and $\mu$ is the Lebesgue measure. Let $K$ be a fat Cantor set of positive measure. Then $K$ is compact and totally disconnected. In particular, $K^{c}$ is dense in $\mathbb{R}^{1}$. I think setting $\epsilon$ to be anything less than $m(K)$ gives us a problem here if we are trying to approximate $\chi_{K}$.
On a similar note, any lower semi-continuous function that is bounded above by $\chi_{K}$ will also be bounded above by the zero function. Hence, if $v$ is such a function, we will always have $ \int_{\mathbb{R}}(\chi_{K}-v)\,dm\geq m(K). $ However, in his statement of the Vitali-Caratheodory theorem, he states that any $L^{1}$ function can be approximated from below arbitrarily close by lower semi-continuous functions. The function $\chi_{K}$ is obviously in $L^{1}$, so what is going wrong here?