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 function

$$\mu \,\colon\, L \longrightarrow [0, \infty]$$

to the non-negative lower reals satisfying:

1. strictness, the bottom element is sent to zero:

   $\mu(\bot) = 0$

2. additivity:

   $\forall_{a,b\in L} \;\;:\;\; (a \wedge b = \bot) \implies \big( \mu(a \vee b) = \mu(a) + \mu(b) \big) \mathrlap{\,.}$

A content that moreover satisfies the conditions

1. modularity:

   $\mu(a) + \mu(b) = \mu(a \vee b) + \mu(a \wedge b)$

2. monotonicity:

   $(a \leq b) \Rightarrow \big(\mu(a) \leq \mu(b)\big)$

is called a valuation.

See also

• valuation (measure theory)

• sigma-continuous valuation

• probability content?

• measure

• Jordan content

References

• Wikipedia, Content (measure theory)