Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 23J / c

No. On the real line, the disjoint bounded Borel sets $[n,n+1)$, $n\ge 0$, each have assigned mass zero, while their union $[0,\infty )$ has assigned mass one. This contradicts countable additivity, so the set function is not a Borel measure.