TITLE: The universal algebra of a $\sigma$-algebra
QUESTION [6 upvotes]: I am searching for the 'dual' algebraic structure of a $\sigma$-algebra. The notion of duality is like in the case of the Boolean algebra and set algebra.
If $X$ is a set, the complement and intersection on the power set of $X$ is called a set algebra and the series of equations that define a Boolean algebra is the dual of this structure. 
I found this link that seems related to my question:
Is there such a thing as the sigma-completion of a Boolean algebra?
but still it does not solve my problem.

REPLY [4 votes]: In the paper [1], Sikorski constructs a duality that generalizes Stone duality to certain $\sigma$-complete Boolean algebras and more generally certain $\kappa$-complete Boolean algebras. I shall outline the duality mentioned in Sikorski's paper here.
Suppose that $\lambda$ is a cardinal. Then a Boolean algebra $B$ is said to be $\lambda$-complete if the least upper bound $\bigvee R$ exists whenever $|R|<\lambda$. A filter $Z$ on a $\lambda$-complete Boolean algebra $B$ is said to be a $\lambda$-complete filter if whenever $|R|<\lambda$ and $R\subseteq Z$, then $\bigwedge R\in Z$ as well. We shall call a $\lambda$-complete Boolean algebra strongly $\lambda$-representable if every $\lambda$-complete filter can be extended to a $\lambda$-complete ultrafilter.
A $P_{\lambda}$-space is a completely regular space such that the intersection of less than $\lambda$ many open sets is open, and a topological space $X$ is said to be $\lambda$-compact if every open cover of $X$ has a subcover of cardinality less than $\lambda$. Sikorski gave a correspondence between all $\lambda$-compact $P_{\lambda}$-spaces and all $\lambda$-representable $\lambda$-complete Boolean algebras. The proof of this result is exactly the same as the proof of the duality between Boolean algebras and compact totally disconnected spaces.

Sikorski, Roman
Remarks on some topological spaces of high power. 
Fund. Math. 37, (1950). 125–136.