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.