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) 



Is there a probability theory developed in intuitionistic logic?
The title of the question is a bit of a misnomer, or at least has the potential to cause confision. "Intutionistic probability theory" means to me "theory of probability developed in intuitionistic logic". But you seem to be asking whether we can replace $\sigma$-algerbas (which are Boolean algebras) with Hetying algebras.


Regarding your first question, the relevant notion is that of a [valuation](http://en.wikipedia.org/wiki/Valuation_%28measure_theory%29). This is like a real-valued measure, but it is defined only on the open sets instead of Borel sets. The open sets form a complete Heyting algebra, so this should provide you with a starting point.


Regarding your second question about Heyting-algebra-valued measures on Borel sets, I would just like to observe that a homomorphism $f : \mathcal{B}(\mathbb{R}) \to \mathcal{H}$ preserves complements (because it preserves $0$, $1$, $\land$ and $\lor$), which means that it factors through the regularization of $\mathcal{H}$ (the Boolean algebra consisting of those elements of $\mathcal{H}$ that are closed under double complement). So you might as well replace $\mathcal{H}$ with its Boolean algebra of regular elements, or else consider instead $\mathcal{H}$-valued valuations $\mathcal{O}(\mathbb{R}) \to \mathcal{H}$.


