Sub-additivity of a measure, basic definition of a measure Here the definition of a measure as given in my lecture notes:
A map $ \mu : 2^X \rightarrow [0, \infty] $ is called a measure on X if:
$1. \ \mu(\emptyset) = 0$
$ 2. \ \mu(A) \leq \sum_{i=1}^{\infty} \mu(A_i) $ if $A \subset \bigcup\limits_{i=1}^\infty A_n$ 
Now, according to my script, 2. implies $\sigma$-subadditivity, i.e.:
$ \mu(\bigcup \limits_{i=1}^\infty A_i) \leq \sum_{i=1}^\infty \mu(A_i)        $
My first question is how to explain this implication. My second question is if I understand it correctly that for the case of ifinity many subsets as in 2., the sets don't have to be disjoint. And my last question would be if condition 1. $\mu(\emptyset) = 0$ is not redundant since it follows from the sigma-subadditivity?
Thanks
 A: As mentioned in a comment: The lecture notes use the non-standard terminology characteristic of Evans-Gariepy's book Measure theory and fine properties of functions. This means: what is called a measure here is usually called an outer measure. Compare with Struwe's Bemerkung 1.1.5 on page 4. For future questions, you should expect that people will be taken aback by this uncommon usage. I'm not aware of any other introductory text to real analysis and measure theory following Evans and Gariepy's conventions.

For your first question you can simply take $A = \bigcup_{i=1}^\infty A_i$ to conclude
$$
\mu(A) = \mu\left(\bigcup_{i=1}^\infty A_i\right) \leq \sum_{i=1}^\infty \mu(A_i)
$$
from condition 2.
Notice that in these lecture notes $A \subset B$ really is $A \subseteq B$, meaning that equality $A = B$ is allowed.
Your interpretation is correct: there is no assumption on (pairwise) disjointness of the sets $A_i$.
For your last question, the condition $\mu(\emptyset)  = 0$ is not redundant, because $\mu\colon 2^X \to [0,\infty]$ given by $\mu(A) = 1$ for all $A \subseteq X$ satisfies condition 2.
