# Lattice property of the class of signed measures

Signed measures have values either in $(-\infty,+\infty]$ or $[-\infty,-\infty)$, to avoid the possibility of adding $+\infty$ to $-\infty$. If $(X,\mathcal{X},\mu)$ is a signed measure space and $A$ is a measurable set, define

$\displaystyle \mu_+(A):=\sup_{B\subset A}\mu(B),\quad \mu_-(A):=-\inf_{B\subset A}\mu(B),\quad |\mu|=\mu_++\mu_-.$

The set function $\mu_-,\mu_+$ and $|\mu|$ are respectively the positive, negative and total variations of $\mu$.

Theorem If $\mu$ and $\nu$ are signed measures on a measurable space, there is a signed measure $\mu\vee \nu$ majorizing $\mu$ and $\nu$ and majorized by every other signed measure majorant $\mu$ and $\nu$.

Proof If $\mu-\nu$ is a well-defined signed measure, that is if $\mu(X)$ and $\nu(X)$ are not both $+\infty$ or both $-\infty$. Let $X_+$ be a maximal positivity set and $X_-$ be a maximal negativity set, for $\mu-\nu$. Define

$\displaystyle (\mu\vee\nu)(A):=\mu(A\cap X_+)+\nu(A\cap X_-).$

This sum defines a measure with the required properties.