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I'm studying about some qualities for sequences of measure functions,and I have this problem:

let $\mu$ be Dirac measure on the X= {$\frac{1}{n}; n\geq 1$} at the point $\frac{1}{i}$, I know that {$\mu_n$} is convergent pointwise, but I guess that the convergent measure is not a measure, Now my question is:

1- is my guess right?

2- How can I prove if {$\mu_n$} is a sequence of finite measures on the measurable space of (X,M) that is uniformly convergent to a finite measure $\mu$, then $\mu$ is a measure on $(X,M)$.

Any help would be great thanks.

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    What do you mean by "convergent pointwise"? Weak-$*$ convergent in $C(X)^*$? What do you mean by "uniformly convergent"? – 2017-01-20

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You can prove measure equality first for finite condition then use finiteness of m for monotonicity and continuity from below . Use the fact that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum . Now two inequality will prove easily .