Maharam algebra

A continuous submeasure or Maharam submeasure on a Boolean algebra is a real-valued function m such that

• ${\displaystyle m(0)=0,m(1)=1,}$ and ${\displaystyle m(x)>0}$ if ${\displaystyle x\neq 0}$.
• If ${\displaystyle x\leq y}$, then ${\displaystyle m(x)\leq m(y)}$.
• ${\displaystyle m(x\vee y)\leq m(x)+m(y)-m(x\wedge y)}$.
• If ${\displaystyle x_{n}}$ is a decreasing sequence with greatest lower bound 0, then the sequence ${\displaystyle m(x_{n})}$ has limit 0.

A Maharam algebra is a complete Boolean algebra with a continuous submeasure.