Why there exists no measure space $(X,\mathcal S,\mu)$ such that $\{\mu(E):E\in \mathcal S\}=[0,1)$.

Question

I am a graduate student of Mathematics.In the book Measure,Integration and Real Analysis by Sheldon Axler there is a question that asks the reader to show that there exists no measure space $(X,\mathcal S,\mu)$ such that $\{\mu(E):E\in \mathcal S\}=[0,1)$.I am not sure how to do it.I was thinking of taking a nested sequence $\{E_n\}$ of sets in $\mathcal S$ such that $\mu(E_n)=1-1/n$ but I don't think that will work.Can someone give me any clue?

Answer 1

Use your idea along with the property of "continuity from below" that all measures have. You can show that an event has measure one in this way.

Answer 2

If there would be a measure space with this property, then for a sequence of sets $(E_n)$ with $E_n\nearrow X$ in $\mathcal{S}$, we have that $\mu(E_n)\leq\mu(E_{n+1})$. Hence we get a monotonic sequence $(\mu(E_n))$ which is bounded and therefor convergent with $\mu(X)=a\in [0,1)$. But then as the measure is monotonic we have $\{\mu(E): E\in\mathcal{S}\}=[0,a]$.

Answer 3

If $(X,\mathcal S,\mu)$ denotes a measure space then $\mathcal S$ denotes a $\sigma$-algebra on $X$ so that by definition $X\in\mathcal S$ and consequently:$$\mu(X)\in\{\mu(E)\mid E\in\mathcal S\}$$ This with $\mu(E)\leq\mu(X)$ for every $E\in\mathcal S$.

So the set $\{\mu(E)\mid E\in\mathcal S\}$ has a largest element in $\mu(X)$.

However evidently the set $[0,1)$ does not have a largest element and we conclude that the two sets do not coincide.