Let [;(A, \mu) ;] be a measure space (A is a sigma-algebra). If [; A_k ;] is a countable collection of sets in [; A ;], prove that [; \mu(\cup_{k=1}^{k=\infty}A_k) \leq \sum_{k=1}^{k=\infty} \mu(A_k) ;].

Can someone break this down into steps? As in, first prove this, then this, then....

## Number154

Are you given countable additivity of disjoint sets as part of the definition? Consider B_k defined as A_k minus the union of all preceding As and show that the measures of the A sets bound the corresponding measures of the B sets.