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.

Advertisements

发表评论

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / 更改 )

Twitter picture

You are commenting using your Twitter account. Log Out / 更改 )

Facebook photo

You are commenting using your Facebook account. Log Out / 更改 )

Google+ photo

You are commenting using your Google+ account. Log Out / 更改 )

Connecting to %s