Let $n \in N$, $(X, A, \mu)$ be a measure space and $E_1 ,. . . , E_n \in A$. Let $n \in N$, $(X, A, \mu)$ be a measure space and $E_1 ,. . . , E_n \in A$. For each $j \in \{1 ,. . . , n\}$ define
$$C_j∶= \{x \in X | x \in E_k \quad \text{for exactly $j$ indices}, k ∈ \{1. . . , n\}\}.$$
I want to find a general expression for all the $C_j$ since I need to prove that each $C_j$ is a measurable set, I made an example and it is as follows
With $n = 3$
$$C_1 = (E_1\cap E_2^c \cap E_3^c)\cup (E_1^c\cap E_2 \cap E_3^c)\cup(E_1^c\cap E_2^c \cap E_3)$$
$$C_2 =(E_1\cap E_2 \cap E_3^c)\cup (E_1\cap E_2^c \cap E_3)\cup (E_1^c\cap E_2^c \cap E_3)$$
$$C_3 = \bigcap_{k=1}^3 E_k$$
So for $C_n$ we already have a general expression:
$$C_n = \bigcap_{k=1}^n E_k$$
for $C_1$:
$$C_1=\bigcup_{j=1}^n \left(E_j \cap \bigcap_{k\neq j}^n E_k^c\right) $$
I was thinking of introducing permutations but actually I don't know if it works, someone can guide me a bit to express the $ C_j $ sets in a manageable way
 A: You can just write:
$$
C_j = \bigcup_{S \subseteq \{1,\dots,n\}, |S| = j} \left(\bigcap_{i \in S} E_i \cap \bigcap_{i \notin S} E_i^c\right)
$$
These may not be "standard" notations, but all that matters is that all unions/intersections here are finite, so it remains measurable.
A: You're already on your way there! Here's a quick hint that should help you wrap things up:
There's no rules over what you can use as an index set for your unions/intersections. I think you're struggling because you want to write them as $\bigcup_{k=1}^n$, but you can index over basically any set you want! Since we want to preserve measurability, we'll need the set to be countable, but other than that the world is your oyster.
So, for instance, there's nothing stopping you from using the set
$$I_3 = \big \{ (A, B) \mid |A| = 3, A \cap B = \emptyset, A \cup B = \{1, \ldots, n \} \}.$$
Then $I_3$ is finite (do you see why?), so we can write
$$C_3 = \bigcup_{(A,B) \in I_3} \left ( \bigcap_{k \in A} E_k \cap \bigcap_{k \in B} E_k^c \right ).$$
Do you see why this gives you exactly the $C_3$ that you want? Moreover, do you see how to generalize this to get $C_j$ for any $j \leq n$?
As an aside, you can do this with permutations as well, and again by indexing over sets more complicated than $\{1 \ldots n \}$ you can write that idea quite cleanly too. For instance, consider
$$C_3 = \bigcup_{\sigma \in \mathfrak{S}_n} \left ( \bigcap_{k=1}^3 E_{\sigma k} \cap \bigcap_{k=4}^n E_{\sigma k}^c \right ).$$
(Here $\mathfrak{S}_n$ is the set of permutations of $n$ elements)
Again, do you see why this works? Do you see how to generalize it to get $C_k$ for $k \neq 3$?

I hope this helps ^_^
