On (1), it seems reasonable to check the conditions in the definition of a measure. However, I'm not exactly sure how to help you, because I don't know what kinds of assumptions you are putting on the function . Are you taking ? Is assumed to be nonnegative?
On (2), you're not quite done. You need to show that given , you can produce so that its measure is greater than . Try using finite / countable additivity with the set that you found.