combined experiments consisting sub-experiments which became dependent on each other The following is stated in the Papoulis' book   (p. 48)

If we have two probability spaces ${(Ω_1, F_1, P_1)}$ and ${(Ω_2, F_2, P_2)}$,
whether  these two experiments are independent or dependent, the
probability space of the combined experiment, which consists of those
two experiments, will be ${(Ω_1 × Ω_2, F, P)}$. To complete our new
probability model, it is sufficient to assign the values of the
probability function P as follows:
$${P(A×Ω_2 )=P_1 (A) \qquad\qquad\qquad\qquad(1)}$$
$${P(Ω_1×B)=P_2 (B)\qquad\qquad\qquad\qquad (2)}$$

But in the following example, where we have a combined experiment consisting of two dependent random experiments, the assignments (1) and (2) leads to a contradiction.

Suppose that we have two types of dice - one is fair, the other is not:

*

*Dice $A$ with probability space ${(Ω_1, F_1, P_1)}$

*

*${A_i}$: The event is that by throwing the dice $A$, the number $i$ will come

*${P_1 (A_6 )=1/6}$



*Dice $B$ with probability space ${(Ω_2, F_2, P_2)}$

*

*${B_i}$: The event is that by throwing the dice $B$, the number $i$ will come

*${P_2 (B_6 )=1/10}$
Our combined experiment is  as follows: In a weird casino, two dice are rolled at the same time and both dice always return the same result.
In this experiment, The probability space will be ${(Ω_1 × Ω_2, F, P)}$ and :
$${P(A_1×B_6 )=P(A_2×B_6 )=⋯=P(A_5×B_6 )=0}$$
$${P(A_6×B_1 )=P(A_6×B_2 )=⋯=P(A_6×B_5 )=0}$$
According to relations (1) and (2):
$${P(A_6×Ω_2 )=P_1 (A_6 )=1/6}$$
$${P(Ω_1×B_6 )=P_2 (B_6 )=1/10}$$
(notice that $P(Ω_1×B_6 ) \neq {P(A_6×Ω_2 )}$)
According to the law of total probability:
$${P(Ω_1×B_6 )=P(A_1×B_6 )+P(A_2×B_6 )+⋯+P(A_6×B_6 )}$$
$${P(A_6×Ω_2 )=P(A_6×B_1 )+P(A_6×B_2 )+⋯+P(A_6×B_6 )}$$
So we have:
$${P(Ω_1×B_6 )=P(A_6×B_6 )}$$
$${P(A_6×Ω_2 )=P(A_6×B_6 )}$$
That is, we have:
$${P(Ω_1×B_6 )=P(A_6×Ω_2 )=1/6}$$
Here we encounter a contradiction:

*

*Papoulis assumptions say:     $\qquad\qquad{P(Ω_1×B_6 )≠P(A_6×Ω_2 ) }$

*the Kolmogorov axioms compel:      $\;\;\quad{P(Ω_1×B_6 )=P(A_6×Ω_2 )}$

now If we want to summarize:
We used Papoulis' assumptions (relations $(1)$ and $(2)$), but by applying the axioms of probability, I finally came to conflicting conclusions with our assumptions.
My question is: Is it really correct to apply assignments $(1)$ and $(2)$ if the sub-experiments of a combined experiment are dependent? (However, my calculations in this example show that this is not good practice)
 A: Too long for a comment.
Papoulu's definition is not quite complete, or right for that matter. He is simply trying to say (in a rather unreadable manner) that a probability space for a combination of two experiments is the Cartesian product and the probability measure in this space is any measure with marginal probability function those of the individual experiments. This wrong in the sense that this is far from specifying a probability space, indeed, there may be infinitely many probability spaces where conditions (1) and (2) occur and both conditions are not enough to specify a probability measure.
In regards to your example, experiments 1 and 2 are incompatible to be combined as requested. Let $X$ the result of the first experiment and $Y$ that of the second. You are stating that $X$ and $Y$ always show the same number, this means that $X = Y.$ However, $P(X = 6) = 1/6$ while $P(Y = 6) = 1/10,$ which is clearly impossible. I have no idea why you brought Kolmogorov's axioms at the end, they don't compell anything, they simply state what are the "axioms" defining probability measures. Anyway, this is the problem of Papoulu's "definition", you have $X$ and $Y$ but it gives you no real way to combine them really. The most common way is to make them independent, that means $P(A_i \times B_j) = P(A_i) P(B_j),$ but there may be many more ways to combine them such that (1) and (2) hold (all these other ways will give rise to not independent experiments). As a side note, you should avoid the term "dependent" since it is not quite clear what that means, it can mean "correlation not zero" or "one is function of the other" or "not independent", these three properties are not as related as intuituion would suggest (e.g. there are random variables $X$ and functions $f(X)$ such that the correlation between $X$ and $f(X)$ is zero; or there are random variables $X = f(Y, Z)$ such that $X$ and $Z$ are independent, etc.).
