For $(X,\mathcal{F})$ a measure space, I know that if we have $\mu\_{n}(A) \searrow$, i.e. is a decreasing sequence of measures for each $A \in \mathcal{F}$ and $\mu\_{1}(X) < \infty$ then $\mu = \lim\_{n \rightarrow \infty} \mu\_{n}$ is not a measure.

The problem asks for a counterexample. I am struggling with coming up with a counterexample for this and would appreciate some help. This problem comes from an early chapter in the book before any discussion of Lebesgue measure, so it should be possible to come up with a counterexample using only the measures discussed at that point which are the counting measure or Dirac measure and linear combinations of measures.

A hint would be appreciated. 

Edit: I am starting to think that this may actually be a measure if $\mu\_{1}(X) < \infty$. Is this the case ?