Conditions for measure being finite and $sigma$-finite

working on a problem for measure theory and wanted to see if I'm on the right track.

I've shown that over a measurable space $(X,\mathcal M)$, $\sum_{j=1}^\infty a_j\mu_j$ where {$\mu_j$}$_{j=1}^\infty$ a sequence of measures, and {$a_j$}$_{j=1}^\infty \subset (0,\infty)$ is a measure.

Now I'm trying to find the conditions for which a specific example of this measure is finite, and $\sigma$-finite, the measure being $$\sum_{j=1}^\infty a_j\delta_{x_j}$$ where $\delta_x$ is the Dirac measure.

$\sum_{j=1}^\infty a_j\delta_{x_j}(X)$ = $\sum_{j=1}^\infty a_j$ since each $\delta_{x_j}$ = $1$ over the entire set X, and condition for finiteness of this measure should be the sum being finite. Is this correct?

Secondly, for $\sigma$-finiteness, my intuition tells me that X should be countable, since the Dirac measure is similar to the counting measure, but I'm not sure how to formally show conditions for finiteness or $\sigma$-finiteness.

Any help would be great! Thank you.

Answer

The finiteness part is correct. The measure is always $\sigma$-finite. Each $x_j$ has finite measure (namely $a_j$) and $X = (\cup_j \{x_j\})\cup (X\setminus \cup_j \{x_j\})$ where $X\setminus \cup_j \{x_j\}$ is measurable with measure $0$.