Two Real Analysis Q's
How would one show the following:
1. If , is a measure iff is continuous from above (i.e. if and then ).
( ) I managed this direction, it's the ( ) that I need help with.
2. If is semifinite and , for there exists with .
Ok, so by def of semifinite we know that ther exists an such that . Could I just set ?
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.
On (1): Yes, I'm assuming that is nonnegative and that not .
is the meausrable space
On (2): I'm not quite sure what you mean. In order to use finite/countable additivity I need disjoint sets, and in order to form these I need at least two sets... however I only have .