It's more complicated than that. In general, there is no universally dominating measure (on countable spaces, the counting measure dominates all finite measures, so sometimes there is).
In this comment:
Take a maximal family $\mathcal{A}$ of mutually singular probability measures. (Use Zorn's Lemma.) The space of measures is isometrically the $l_1$-sum of $L^1(\nu)$ as $\nu$ ranges over the family $\mathcal{A}$.
GEdgar, the author of the answer, explains the construction.
Since you used $\mathcal{A}$ for the $\sigma$-algebra, let us call the maximal family of mutually singular probability measures $\mathcal{P}$.
Now if $\mu$ is a finite (signed) measure on $\mathcal{A}$, for every $\nu \in \mathcal{P}$ you have the Lebesgue decomposition
$$\mu = \mu_{\nu} + \mu_{\nu}^{\perp}$$
of $\mu$ with respect to $\nu$, where $\mu_{\nu}$ is absolutely continuous with respect to $\nu$ and $\mu_{\nu}^{\perp}$ and $\nu$ are mutually singular. Since the members of $\mathcal{P}$ are mutually singular, $\mu_{\nu_1}$ and $\mu_{\nu_2}$ are mutually singular for $\nu_1\neq \nu_2$.
Then you have
$$\mu = \sum_{\nu \in \mathcal{P}} \mu_{\nu},$$
for otherwise $\mu$ would have a nonzero part that is singular with respect to all $\nu \in \mathcal{P}$, and from that you could construct a probability measure (the positive or negative part of the remainder would be nonzero, and after scaling gives a probability measure) that is singular with respect to all $\nu\in \mathcal{P}$, contradicting the maximality of $\mathcal{P}$.
And identifying $\mu_{\nu}$ with its density with respect to $\nu$ we have the map sending each measure into the space
$$\bigoplus_{\nu \in \mathcal{P}}{}^{l_1} L^1(\nu),\tag{1}$$
the $l_1$-direct sum of all the $L^1(\nu)$. One then verifies that this map is a bijective isometry.
We can identify the space $(1)$ with $L^1(\mu)$ for some measure $\mu$, but that can in general not be a measure on $\mathcal{A}$. The obvious construction is to let $\mathcal{Y} = \mathcal{X}\times \mathcal{P}$, and take the $\sigma$-algebra
$$\mathcal{B} = \bigl\{ M \subset \mathcal{Y} : \bigl(\forall \nu \in \mathcal{P}\bigr)\bigl( \{ x \in \mathcal{X} : (x,\nu) \in M\} \in \mathcal{A}\bigr)\bigr\}$$
and set
$$\mu(M) = \sum_{\nu \in \mathcal{P}} \nu\bigl(\{ x\in \mathcal{X} : (x,\nu) \in M\}\bigr)$$
for $M \in \mathcal{B}$. Sometimes, smaller constructions are possible.
