I was reading the Measure (mathematics) page on wikipedia:
http://en.wikipedia.org/wiki/Measure_%28mathematics%29
I was confused with one of their sections at the end, "Additivity". They were strengthening the condition of measure, countable additive. They say for any set $I$ and any set of non-negative $r_i, i \in I,$ the sum of $r_i$ is defined to be supremum of all the sums of finitely many of them.
In this case is $r_i$ a set, or a number? If it is a number, why do we need the supremum? If it is a set, what do they mean by a non-negative set, or how to sum sets? Lastly, how does this strengthen countable additive condition of measure, does it weaken the disjoint condition? I had a similar confusion about the second statement they have in that same section, are $\lambda$ and $\kappa$ numbers or sets? What do they mean by $\lambda$ < $\kappa$, if they are sets? If possible, simple examples would help.