Range of measure function cannot be half open interval

Question

Explain why there does not exist a measure space $(X,S,\mu)$ with the property that $\{ \mu(E):E \in S \}= [0, 1)$

My approach was the following -

For every $n \in \mathbb{N}$, there is a set $E_n \subset X$ such that $\mu (E_n) = \frac{1}{2^n}$

If these $E_i$'s were disjoint, we could say that $\mu (\cup E_n) = \sum \mu(E_n) = 1$ which is a contradiction. However, I can't say that the $E_i$'s are disjoint.

Answer 1

Let's assume that there exists such a measure and set $a:= \mu(X)\in [0,1)$, . Then $\mu(A)\leq a$, for all $A\subset X$. Since $\{ \mu(E):E \in S \}= [0, 1)$, for every $n$ there exist $A_n\subset X$ such that $\mu(A_n)=1-\frac{1}{n}$ and for large enough $n$ this is bigger than $a$, which is a contradiction.

Answer 2

We definitely want to look at a countable union of sets. Let's just not bother with disjointness. For each $n$, let $E_{n}$ have measure $1 - \frac{1}{n}$. What's the measure of $\cup_{n = 1}^{\infty} E_{n}$?