Is there a standard name for this probability-theoretic construction? What follows is a description of a concept in probability theory that I find very useful, but cannot find described in the literature. I've been calling it the "perplectic sum", but that's a term I made up. The idea seems fundamental enough that I can't be the first one to have thought of it, and I'd like to know if it has a standard name and/or notation.
Let $A = (\Omega_A, \mathcal F_A, P_A)$, $B = (\Omega_B, \mathcal F_B, P_B)$ be two probability spaces. Let $PP$ denote perplexity, i.e., $PP(A) = e^{H(A)}$ where $H(A)$ is the entropy of $A$ measured in nats. Then the perplectic sum $(\Omega, \mathcal F, P) = A \oplus_{PP} B$ is defined as follows:


*

*$\Omega$ = $\Omega_A \sqcup \Omega_B$, i.e. the disjoint union (categorical coproduct) of $\Omega_A$ and $\Omega_B$.

*$\mathcal F = \mathcal F_A \times \mathcal F_B$.

*$P((X,Y))$ = $P_A(X) \cdot \frac{PP(A)}{PP(A)+PP(B)}$ + $P_B(Y) \cdot \frac{PP(B)}{PP(A)+PP(B)}$.
Trivial example: if $P_A$ is defined by the discrete uniform distribution over the set $\{1,2\}$ and $P_B$ is defined by the discrete uniform distribution over the set $\{3,4,5,6\}$, then $P$ is defined by the discrete uniform distribution over the set $\{1,2,3,4,5,6\}$.
 A: The object you have in mind is the probability space $(\Omega,\mathcal F,P)$ defined by $\Omega=\Omega_A\cup\Omega_B$ under the condition that $\Omega_A\cap\Omega_B=\varnothing$, by $\mathcal F=\{C\cup D\mid C\in\mathcal F_A,D\in\mathcal F_B\}$ (hence, definitely not the definition that you wrote), and by $P(C\cup D)=tP_A(C)+(1-t)P_B(D)$ for every $C$ in $\mathcal F_A$ and $D$ in $\mathcal F_B$.
To make sense of this definition, one should note that each $F$ in $\mathcal F$ can be written uniquely as $F=C\cup D$ with $C$ in $\mathcal F_A$ and $D$ in $\mathcal F_B$ (namely, consider $C=F\cap\Omega_A$ and $D=F\cap\Omega_B$).
In the "Trivial example" at the end of your post, this yields any probability measure $P$ on $\Omega=\{1,2,3,4,5,6\}$ such that $P(\{1\})=P(\{2\})$ and such that $P(\{3\})=P(\{4\})=P(\{5\})=P(\{6\})$ (but not necessarily the uniform distribution).
My guess is that most people, when faced with this construction and asked for a name, would call it (admittedly rather loosely) a disjoint union of probability spaces (and they would explain the meaning of the phrase right away).
Finally, a caveat is in order at this point: an example or two of results which the constructions in the paper you mention in  comment, allow to reach, would be useful to estimate the worthiness of the whole enterprise (the paper in question (admittedly preliminary) seems to give none).
