Bernoulli Measure and Joint PDF: Are these Wiki definitions correct? I have a question on the Bernoulli Measure and Joint PDF.  Using the Wiki definitions my results are not consistent.  I first present my understanding with definitions and then ask questions.
Wiki defines the Bernoulli Measure as:
a)
$$
P(X_1 = x_1, X_2 = x_2, ..., X_n = x_n) = p^k(1-p)^{n-k}
$$
where $k$ is # of successes in $n$ trials.  Question this omits the binomial coefficient.  Is it correct?
The above measure is a Joint PDF which Wiki defines as follows:
b)
$$
\rho_{X_1,...,X_n}(x_1, ..., x_n) = P(X_1 = x_1 \land ... \land X_n = x_n) = P(X_1 = x_1) P(X_2 = x_2 | X_1 = x_1) P(X_3 = x_3 | X_1 = x_1, X_2 = x_2) ... P(X_n = x_n | X_1 = x_1, X_2 = X_2, ..., X_{n-1} = x_{n-1})
$$
The RHS above is from application of the chain rule of conditional probability.  Using the following equality $(X = z) = \{s \in \Omega : X(s) = z\}$ it's my understanding this equals:
c)
$$
\rho_{X_1, ..., X_n}(a_1, ... , a_n) = P(X_1 = a_1, ... , X_n = a_n) = P(\{s \in \Omega : X_1(s) = a_1 \land ... \land X_n(s) = a_n\}) = P(\{s \in \Omega : X_1(s) = a_1\} \cap ... \cap \{s \in \Omega : X_n(s) = a_n\}
$$
Question is form (c) correct?
I have doubts on (c).  For example, if there is a Bernoulli Measure with sample space, probabilities and random variable:
d)
$$
X_i : \Omega \rightarrow \mathbb{R}
$$
$$
\Omega = \{\epsilon_0, \epsilon_1\}
$$
$$
X_i(\epsilon_0) = 1, X_i(\epsilon_1) = 0
$$
$$
P(\epsilon_0) = P(X_i = 1) = p, P(\epsilon_1) = P(X_i = 0) = 1 - p
$$
Now considering two steps in the measure:
e)
$$
P(X_1 = 0, X_2 = 1) = p^1 (1 - p)^{2 - 1} = p (1 - p)
$$
but when substituting events from the sample space there is an inconsistency between above and what follows.  This makes me wonder if my definition of the JPDF or Bernoulli measure is incorrect:
f)
$$
P(X_1 = 0, X_2 = 1) = P(\{\epsilon_0\} \cap \{\epsilon_1\}) = P(\{\}) = 0 
$$
 A: "This makes me wonder if my definition of the JPDF or Bernoulli measure is incorrect": They are both correct, except you are talking about joint PMFs, not joint PDFs. In fact, in the Wiki link you shared, the joint PMF is already defined as $$p_{X_!,\dots,X_n}(x_1,\dots,x_n)=P(X_1=x_1\cap\cdots\cap X_n=x_n)$$
in Eq.4, which is exactly the same as (c) (maybe you missed this definition, it's right above equation (b), which you wrote above, in the link). (b) is just an equivalent form and you will see that form (c) is much more common than (b).
For the Bernoulli variables, the problem lies in your sample space $\Omega$ and the probability space in general. The probability space $(\Omega,\mathcal F:=2^\Omega,P)$ defined in your question makes sense only if you consider a single Bernoulli trial. If we have a sequence $\{X_i\}_{i=1,\dots,n}$ of $n$ i.i.d. Bernoulli trials, the sample space, say $\Omega'$, should be the product space $\Omega'=\{\epsilon_0,\epsilon_1\}^n$. In our case, $n=2$ and thus, $\Omega'=\{\epsilon_0,\epsilon_1\}\times \{\epsilon_0,\epsilon_1\}=\{(\epsilon_0,\epsilon_0),(\epsilon_0,\epsilon_1),(\epsilon_1,\epsilon_0),(\epsilon_1,\epsilon_1)\}.$
So, the probability space is the triple $(\Omega',\mathcal F',P'),$ where  $\mathcal F':=\mathcal F\otimes \mathcal F$ (tensor product) and $P':=P\times P$ is the product measure.
Considering this probability space, we have \begin{align} 
P'(X_1 = 0, X_2 = 1)&=P'(\{(\epsilon_1,\epsilon_0),(\epsilon_1,\epsilon_1)\}\cap \{(\epsilon_0,\epsilon_0),(\epsilon_1,\epsilon_0)\})\\ &=P'(\{(\epsilon_1,\epsilon_0)\})\\&=P(\epsilon_1)P(\epsilon_0)=(1-p)p
\end{align}
as required.
