nLab content (measure theory)

Contents

This entry is about the notion of “content” in measure theory. For the notion in ring theory, see content (ring theory) and for the notion in combinatorics/representation theory set at hook-content formula. For the contents sidebar of this wiki, see contents. For more disambiguation see content.

Contents

Definition

In measure theory, a content on a distributive lattice (L,,,,,)(L, \leq, \bot, \vee, \top, \wedge) is a valuation μ:L[0,]\mu:L \to [0, \infty] to the non-negative lower reals such that the valuation of the bottom element is zero and additive if the pair of elements is disjoint.

aL.bL.μ()=0\forall a\in L. \forall b \in L. \mu(\bot) = 0
aL.bL.(ab=)(μ(ab)=μ(a)+μ(b))\forall a\in L. \forall b \in L. (a \wedge b = \bot) \implies (\mu(a \vee b) = \mu(a) + \mu(b))

See also

References

Last revised on June 6, 2022 at 03:49:49. See the history of this page for a list of all contributions to it.