Are these terms "Finite and Countable Monotonicity" generally used ?
I see them defined in http://mathworld.wolfram.com/FiniteMonotonicity.html and http://mathworld.wolfram.com/CountableMonotonicity.html as well as in Royden and Fitzpatrick - Real Analysis p. 339. For example, "for any countable collection of sets $\{E_i\}$ that cover a measurable set $E$ then $\mu(E) \le \Sigma \mu(E_i)$. The wolfram definitions refer to the R&F ones, so does anyone else use them ?
Most sources I can refer to seem to focus on monotonicity (as in $A \subset B \implies \mu (A) \le \mu (B)$ ) and sub-additivity, where sub-additivity merely refers to a union, $\mu(\cup E_i) \le \Sigma \mu(E_i)$
Furthermore, these definitions for finite and countable monotonicity don't seem to follow from the generally accepted definition of monotonicity. Indeed, there is an article, http://www.math.wustl.edu/~sawyer/handouts/MeasuresSemiRings.pdf p.3 which gives a result far more deserving of the title "finite monotonicity", namely that if $\{A_i\}_{i=1, n} $ are disjoint and $\cup _{i=1, n} A_i \subset A$ then $\Sigma_{i=1, n} \mu(A_i) \le \mu(A)$