measures 14

We defined an inner measure $m^i$ for an arbitrary set $C$;
$$m^i(C)=|J|-m^o(J\cap C^c) .$$
, where $m^o$ was an outer measure, and $J$ was a measurable set which covered $C$ fully.

An outer measure will be also defined
$$m^o(C)=\inf \left\{\sum m(J_i)| C\subset\cup J_i  \right\} .$$

You may think an inner measure defined by
$$m^{ii}(C)=\sup\left\{ \sum m(I_i)| \cup I_i\subset C   \right\} .$$

However, $m^i$  can handle more sets than $m^{ii}$ .

For example, given $\Omega=[0,1]$ and
$$f(x)=\left\{ \begin{array}{ll} x=1 & x\in\mathbb{Q} \\ x=0 & x\notin\mathbb{Q} \end{array}    \right.$$

What does the measure of $f$ ?