First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this.

Let $X$ be a set of uncountable cardinality. Using the Principle of Well Order we have a well ordering $\le$ on X(and $<$ means $\le$ but not euqual). By adding an element, which we denote by $∞$, and introducing the convention that $x < ∞$ for all $x ∈ X$, we will assume that X has a maximum with respect to $\le$.

We define $ω_1 = \text{min} \{ x ∈ X : \{y ∈ X : y < x \} \ \text{is uncountable}\} $

Clearly such a $ω_1$ exists. we define $0 = \text{min} \ X$ and intervels $[0,x], [x,y), (a,b]$ etc. in the usual sense.

Here are my questions:

(1)For any countable $A ⊂ [0, ω_1)$ there is an $x < ω_1$ so that $A ⊂ [0, x]$.(I don't know how to make use of the countability and uncountablity here)

(2)Equipped with the topology generated by open intervals, $[0, ω_1]$ is compact.

(3)A famous application of the first uncountable ordinal is to find an example of a Borel measure (with respect to the topology in (2)) that is not Radon. So how to construct a finite Borel measure $µ$ on $[0, ω_1]$ which is not a Radon measure?

Any solutions or elementary references will be appreciated!

Solution 1:

For the first problem, observe that $[0,a]$ is countable for each $a\in A$. Let $B=\bigcup_{a\in A}[0,a]$.

• Show that $B$ is countable.
• Show that for each $b\in B$, $[0,b]\subseteq B$.
• Conclude that $B$ is a countable initial segment of $[0,\omega_1)$ and therefore cannot be all of $[0,\omega_1)$.

For the second problem, observe that it suffices to show that every open cover of $[0,\omega_1]$ by open intervals has a finite subcover. Let $\mathscr{U}$ be such an open cover. There is some $U_0\in\mathscr{U}$ such that $\omega_1\in U_0$, so $U_0=(a_0,\omega_1]$ for some $a_0<\omega_1$. Now argue that there must be some $U_1\in\mathscr{U}$ containing $a_0$, so $U_1=(a_1,b_1)$ for some $a_1<a_0$ and $b_1>a_0$. Continuing in this manner, you get a strictly decreasing sequence $\langle a_0,a_1,\ldots\rangle$. Now use the fact that the ordering of $[0,\omega_1]$ that we’re considering is a well-order.

For the third question, consider the $\{0,1\}$-valued measure that assigns measure $1$ to a Borel set $B$ if and only if $B$ contains an uncountable, closed subset of $[0,\omega_1]$.