# How to define measurability for general lattices?

Smiley defined measurability as follows:

(1) Le $L$ be a lattice and $\mu$ be a real-valued function on $L$. An element is called $\mu$-measurable if, for every $b\in L$, $\mu(a\vee b) + \mu(a\wedge b) = \mu(a) +\mu(b)$. Let $L(\mu)$ denote the set of all these $\mu$-measurable elements.

Smiley further proved that, if $L$ is modular, then $L(\mu)$ is a sublattice of $L$.

This definition of measurability is not general enough. It seems that modularity is also a necessary condition for $L(\mu)$ to be a sublattice. Moreover, in order to make $L(\mu)$ to be distributive, $L$ must be distributive too. As Birkhoff has shown, probability functions can only defined on distributive lattices.

Smiley made some other generalizations later but all of these generalizations assume modularity for lattices. So they are not general enough. My question is: How to define a general notion of measurability for general lattices?

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