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 $f$ ?

measures 13

Supposed that a set $C$ , and a set $J$  which fully covers $C$ .
$C\subset J$

The set $C$  is arbitrary.   Its shape may be complicated, or not .

However $J$  is measurable.
For example, $J$  is a big square, and $m(J)$ is height multiplied by width.
$m(C)\leq m(J)$

If a set is measurable, the value of the measure is equal to it of the outer measure. 
$m(J)=m^o(J)=|J|$

Then, an inner measure $m^i$  will be defined by the outer measure,
\[  m^i(C)=|J|-m^o(J\cap C^c) . \] 
, where $C^c$  is the complementary set of $C$ .

We must accept that there exist some measurable sets $J$ which fully cover an aribtrary set $C$ .
It would not be strong opposition.