Let $\mu$ be a measure on a semiring $A \subseteq P(\Omega)$ and let $\mu ^*$ be the outer measure generated by $(A, \mu)$ Prove the semi-inclusion-exclusion principle.
$\mu ^*(E \cup F) + \mu ^*(E \cap F) \leq \mu ^* (E) + \mu ^*(F)$ $E,F \subseteq \Omega$
So far I have proved the following:
$E \cup F = (E \cap F) \cup (E \cap F^c) \cup (F \cap E^c)$
So $\mu ^* (E \cap F) \leq \mu ^* (E \cap F) + \mu ^* (E \cap F^c) + \mu ^* (F \cap E^c)$
Since $E \cap F^c \subseteq E$ we know $\mu ^* (E \cap F^c) \leq \mu ^* (E)$
And since $E^c \cap F \subseteq F$ we know $\mu ^* (E^c \cap F) \leq \mu ^* (F)$
So $\mu ^*(E \cup F) - \mu ^*(E \cap F) \leq \mu ^* (E) + \mu ^*(F)$ $E,F \subseteq \Omega$
How do you get an addition instead of a subtraction?