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)

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)

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.)

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)