Since Boole it is known that probability theory is closely related to logic.   

According to the axioms of Kolmogorov, probability theory is formulated with a (normed) 
probability measure $\mbox{Pr}\colon \Sigma \to [0,1]$ on a Boolean
$\sigma$-algebra $\Sigma$ (of events).


Realizing these data by a set $X$ (sample space of elementary events) and a corresponding $\sigma$-algebra $\Sigma(X)\subseteq P(X)$ of subsets of $X$, one obtains a probability space $(X,\Sigma(X),\mbox{Pr})$.   

 
The $\sigma$-homomorphisms  $f \colon {\cal B}({\mathbb R})\to \Sigma$ (real $\Sigma$-valued measures) are defined on the Borel-$\sigma$-algebra 
${\cal B}({\mathbb R})$ of the real Borel-sets. They can be realized by real-valued measurable functions $F\colon X\to {\mathbb R}$ (random variables). 

I wonder how this theory extends from the classical to the intutionistic logic i.e. from the  Boolean to the Heyting ($\sigma$-) algebras and what  the major differences between the two theories are.

Where can I find precise descriptions of the following topics:

1.  Definition and properties of probability measures on a Heyting algebra ${\cal H}$.

2.  Definition and properties of  real ${\cal H}$-valued measures $f \colon {\cal B}({\mathbb R})\to {\cal H}$.





(Already the discrete case would be of interest.)

(BTW: Boole 1815 - 1864; Heyting 1898 - 1980; Kolmogorov 1903 - 1987)