# Space as a union of disjoint union and intersection

Let $$(X,\mu)$$ and $$E_n\subset X$$ s.t $$\mu(E_n)=\mu(X)$$

can we write $$\mu(X)=\mu(\cap_{n=1}^{\infty}E_n\uplus (\cap_{n=1}^{\infty}E_n)^C)$$?

Shouldn't it be:

$$\mu(X)=\mu(\cup_{n=1}^{\infty}E_n\uplus (\cap_{n=1}^{\infty}E_n)^C)$$?

$$X$$ is the disjoint union of $$\cap E_n$$ and $$(\cap E_n)^{c}$$ so the first one is correct (without any assumptions on the sets $$E_n$$).
• Elementary, I should have looked at it as $X=A\cup A^C$ where $A=\cap E_n$ just as you said Oct 4 '19 at 11:53