A Boolean $\sigma$-algebra is a Boolean algebra $(\mathcal{B},\vee,\wedge,1,0)$ such that every countable collection $(a_n)_{n\in\mathbb{N}}$ has a supremum (for the partial order $a\leq b$ whenever $b=a\vee b$).
A measure on $\mathcal{B}$ could be some function $\mu:\mathcal{B}\rightarrow\mathbb{R}$ which is non-negative, $\mu(0)=0$ and $\sigma$-aditive.
A pair $(\mathcal{B},\mu)$ is called a measure-algebra.
Is there any book where measure theory is studied on abstract measure algebras?
Thanks
The standard reference for measure algebras is volume 3 of David Fremlin's massive treatise on measure theory.
For a dense but shorter introduction, you can read Fremlin's chapter in the third volume of the Handbook of Boolean Algebras, edited by Monk and Bonnet.
Other treatments you might consult are "Boolean Algebras in Analysis" by Vladimirov and "Probability Algebras and Stochastic Spaces" by Kappos.
It should be pointed out that the definition of a measure algebra usually requires the additional condition that the "measure" is positive on nonzero elements.