On the Lukasiewicz probability theory on IF-sets

Abstract

A review of main methods of the probability theory on IF-events is presented in the case that the used connectives are Lukasiewicz
\begin{align}\label{b_1}
    \nonumber
    f\oplus g&=(f+g)\wedge 1\,,\\
    \nonumber
    f\odot g&=(f+g-1)\vee 0\,,
\end{align}
($f$, $g$ are functions, $f,g:\Omega\rightarrow\left\langle 0,1\right\rangle$). Representation theorem for probabilities on IF-events is given. For sequences of independent observables the central limit theorem is presented as well as basic results about conditional expectation. Finally the Lukasiewicz probability theory to the MV-algebra probability theory is embedded.