I will probably get back to you on a purely algebraic combinatorial characterization (i.e. a characterization without using measures) of the random algebra on $2^{\omega}$ tomorrow or so.

1. H.L. Royden, Real analysis, Third edition (1988).

One can also give a purely combinatorial characterization of the random algebra that does not refer to measures. In the paper [2], Thomas Jech gives various characterizations of the Boolean algebras $B$ such that there is a function $\mu:B\rightarrow[0,\infty)$ where $(B,\mu)$ is a measure algebra. We shall therefore add conditions to Jech's characterization of measure algebras that guarantee that the measure algebra is isomorphic to the random algebra.

$\mathbf{Theorem}$ $\mathbf{1}$ A measure algebra $(B,\mu)$ is isomorphic to the random algebra if and only if $B$ is atomless and there is a countable subset $D\subseteq B$ such that for each $b\in B$ there are $R_{n}\subseteq D$ for natural numbers $n$ where $b=\bigwedge_{n}\bigvee R_{n}$.

$\mathbf{Proof}$ $\leftarrow$. Let $\mathcal{B}$ be a countable basis for the topology on $[0,1]$. Then for each Borel set $B$ there is a sequence of open sets $U_{n}$ where $B\subseteq\bigcap_{n}U_{n}$ and $m((\bigcap_{n}U_{n})\setminus B)=0$. Since $\mathcal{B}$ is a basis, for all $n$ there is some $\mathcal{R}_{n}\subseteq\mathcal{B}$ with $U_{n}=\bigcup\mathcal{R}_{n}$. Therefore, the sets $B$ and $\bigcap_{n}\bigcup\mathcal{R}_{n}$ differ only by a set of measure zero. By taking the Borel sets modulo the measure zero sets, we conclude there is a countable subset $D$ of the random algebra $B$ such that for each $b\in B$ there is some $R_{n}\subseteq D$ for all $n$ such that $b=\bigwedge_{n}\bigvee R_{n}$.

$\rightarrow$. Assume that $(B,\mu)$ is an atomless measure algebra such that there is a countable dense subset $D\subseteq B$ where for each $b\in B$ there are $R_{n}\subseteq D$ for natural numbers $n$ where $b=\bigwedge_{n}\bigvee R_{n}$. To show that $(B,\mu)$ is isomorphic to the random algebra, it suffices to prove that $B$ is separable as a metric space. Let $E$ be the Boolean subalgebra of $D$ generated by $B$. Then $E$ is a countable subset of $B$. We claim that $E$ is a dense subset of $B$ as well.

Let $b\in B$. Then there are $R_{n}\subseteq D$ for each $n\geq 0$ such that $b=\bigwedge_{n=0}^{\infty}\bigvee R_{n}$. For all $n\geq 0$, let $S_{n}=\{a_{0}\wedge...\wedge a_{n}|a_{0}\in R_{0},...,a_{n}\in R_{n}\}$. Then $S_{n}\subseteq E$ for all $n$, and $\bigvee S_{n}=(\bigvee R_{0})\wedge...\wedge(\bigvee R_{n})$ for $n\geq 0$. Therefore the sequence $(\bigvee S_{n})_{n}$ is a decreasing sequence with $\bigwedge_{n}\bigvee S_{n}=\bigwedge_{n=0}^{\infty}\bigvee R_{n}=b$. Therefore, we have $^{Lim}_{n\rightarrow\infty}\bigvee S_{n}=b$ in the topology given by the metric defined on any measure algebra. Furthermore, for all $n$, we have $\lim_{T\subseteq S_{n},T\,\textrm{is finite}}\bigvee T=\bigvee S_{n}$, so there is a finite subset $T_{n}\subseteq S_{n}$ such that $d(\bigvee T_{n},\bigvee S_{n})<\frac{1}{n}$. Therefore, we have $^{\lim}_{n\rightarrow\infty}\bigvee T_{n}=b$. However, since $\bigvee T_{n}\in E$ for all $n$, we conclude that $E$ is a dense subset of $B$. We conclude that $(B,\mu)$ is a separable atomic measure algebra, so $(B,\mu)$ is isomorphic to the random algebra. QED

By combining the above characterization of the measure algebras which are isomorphic to the random algebra and Jech's characterizations of the measure algebras, we have the following algebraic characterizations of the random algebra that do not make any reference to measures, but we first need to give some definitions.

A Boolean algebra $B$ is said to be weakly distributive if for every sequence $(p_{n})_{n}$ of partitions of $B$ there is a partition $p$ of $B$ such that for each $a\in p$ the set $\{b\in p_{n}|a\wedge b>0\}$ is finite for each $n$.

A subset $r\subseteq B^{+}=B\setminus\{0\}$ is said to be a cellular family if $a\wedge b=0$ whenever $a,b\in r$ and $a\neq b$ (I should note that Jech calls cellular families antichains in [2]. ).

In a $\sigma$-complete Boolean algebra, we let $\overline{\lim}_{n\rightarrow\infty}a_{n}=\bigwedge_{n=0}^{\infty}\bigvee_{k=n}^{\infty}a_{k}$.`

A $\sigma$-complete Boolean algebra $B$ is said to be uniformly concentrated if there is a function $F$ where $f(c)\in c$ for each cellular family $c$ such that if $c_{n}$ is a finite cellular family with $|c_{n}|\geq 2^{n}$ for all $n$, then $^{\lim}_{n\rightarrow\infty}F(c_{n})=0$.

$\mathbf{Theorem}$ $\mathbf{2}$ Let $B$ be a $\sigma$-complete Boolean algebra. Then the following are equivalent.

I. $B$ is isomorphic to the random algebra.

II. $B$ satisfies the following properties.

$\phantom{spa}$ i. $B$ is atomless.

$\phantom{spa}$ ii. $B$ is weakly distributive

$\phantom{spa}$ iii. there is a countable subset $D\subseteq B$ such that if $b\in B$, then there is a sequence $(R_n)_n$ of subsets of $D$ such that $b=\bigwedge_{n}\bigvee R_{n}$.

$\phantom{spa}$ iv. there is a sequence $(C_{n})_{n}$ of subsets of $B$ with $B^{+}=\bigcup_{n}C_{n}$ and where

$\phantom{spaces}$ a. For all $n$ there is an integer $K(n)$ such that every cellular family in $C_{n}$ has at most $K(n)$ elements.

$\phantom{spaces}$ b. If $a_{n}\not\in C_{n}$ for all $n$, then $\overline{Lim}_{n\rightarrow\infty}a_{n}=0$.

III. The Boolean algebra $B$ satisfies the following.

$\phantom{spa}$ i. $B$ is atomless

$\phantom{spa}$ ii. there is a countable subset $D\subseteq B$ such that if $b\in B$, then there is a sequence $(R_n)_n$ of subsets of $D$ such that $b=\bigwedge_{n}\bigvee R_{n}$.

$\phantom{spa}$ iii. there is a sequence $(C_n)_n$ of subsets of $B$ with $B^{+}=\bigcup_{n}C_{n}$ such that

$\phantom{spaces}$ a. For all $n$ there is an integer $K(n)$ such that every cellular family in $C_{n}$ has at most $K(n)$ elements.

$\phantom{spaces}$ b. If $a_{n}\not\in C_{n}$ for all $n$, then $\overline{Lim}_{n\rightarrow\infty}a_{n}=0$.

$\phantom{spaces}$ c. For all $k$, if $\overline{Lim}_{n\rightarrow\infty}a_{n}=0$, then $a_{n}\in C_{k}$ for all but finitely many $n$.

IV. $B$ satisfies the following.

$\phantom{spa}$ i. $B$ is atomless.

$\phantom{spa}$ ii. there is a countable subset $D\subseteq B$ such that if $b\in B$, then there is a sequence $(R_{n})_{n}$ of subsets of $D$ such that $b=\bigwedge_{n}\bigvee R_{n}$.

$\phantom{spa}$ iii. $B$ is weakly distributive.

$\phantom{spa}$ iv. $B^{+}=\bigcup_{n}C_{n}$ for some sequence $C_{n}$ of sets where

$\phantom{spaces}$ a. For all $n$ there is an integer $K(n)$ such that every cellular family in $C_{n}$ has at most $K(n)$ elements.

$\phantom{spaces}$ b. If $a\vee b\in C_{n}$, then $a\in C_{n+1}$ or $b\in C_{n+1}$.

V. $B$ satisfies the following.

$\phantom{spa}$ i. $B$ is atomless.

$\phantom{spa}$ ii. there is a countable subset $D\subseteq B$ such that if $b\in B$, then there is a sequence $(R_{n})_{n}$ of subsets of $D$ such that $b=\bigwedge_{n}\bigvee R_{n}$.

$\phantom{spa}$ iii. $B$ is weakly distributive.

$\phantom{spa}$ iv. $B$ is uniformly concentrated.

$\mathbf{Proof}$ Simply combine the characterizations of measure algebras in [2] with Theorem 1. QED

We may also use [2] to get similar representations of the measure algebras produced by the standard measure on $2^{\kappa}$ for all cardinals $\kappa$ using Jech's results and Maharam's theorem. In order to state Maharam's theorem we will need a couple definitions.

If $B$ is a Boolean algebra, then recall that $B\upharpoonright a$ is the relative Boolean algebra $\{b\in B|b\leq a\}$.

If $B$ is a complete Boolean algebra, then we say that $X\subseteq B$ completely generates $B$ if whenever $X\subseteq C\subseteq B$ and $C$ is a complete subalgebra of $B$, then $C=B$. Let $\tau(B)=\min\{|X|:X\subseteq B,X\,\textrm{completely generates}\,B\}$.

We say that a complete Boolean algebra $B$ is $\tau$-homogeneous if $\tau(B)=\tau(B\upharpoonright a)$ for each $a\in B^{+}$.

Let $\mu_{\kappa}$ be the standard measure on $2^{\kappa}$. Let $Bor_{\kappa}$ denote the set of Borel subsets of $2^{\kappa}$ and let $I_{\kappa}$ be the Borel subsets of $2^{\kappa}$ of measure zero. Let $P_{\kappa}=Bor_{\kappa}/I_{\kappa}$, and let $\tilde{\mu}_{\kappa}$ denote the measure on the Boolean algebra $P_{\kappa}$ induced by $\mu_{\kappa}$.

$\mathbf{Theorem}$ $\mathbf{3}$(Maharam)(see [3][Ch. 22 Sec. 3]) If $(B,\mu)$ is a $\tau$-homogeneous measure algebra with $\tau(B)=\kappa$ and $\mu(1)=1$. Then $(B,\mu)$ is isomorphic as a measure algebra to $(P_{\kappa},\tilde{\mu}_{\kappa})$.

Therefore, if we combine Maharam's theorem with the characterizations of the measure algebras given by Jech in [2], then we obtain purely algebraic characterizations of the algebras $P_{\kappa}$.

References.

1. H.L. Royden, Real analysis, Third edition (1988).

2. Jech, Thomas. Algebraic characterizations of measure algebras. Proc. Amer. Math. Soc. 136 (2008), no. 4, 1285–1294.

3. Monk, J. Donald, Robert Bonnet, and Sabine Koppelberg. Handbook of Boolean Algebras. Vol. 1-3. Amsterdam: North-Holland, 1989.