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 (normalized) 
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)