nLab content (measure theory)

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.

Definition

In measure theory, a content on a distributive lattice $(L, \leq, \bot, \vee, \top, \wedge)$ is a valuation $\mu:L \to [0, \infty]$ 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.

$$\forall a\in L. \forall b \in L. \mu(\bot) = 0$$

$$\forall a\in L. \forall b \in L. (a \wedge b = \bot) \implies (\mu(a \vee b) = \mu(a) + \mu(b))$$

See also

• valuation (measure theory)

• sigma-continuous valuation

• probability content?

• measure

• Jordan content

References

• Wikipedia, Content (measure theory)