Is there a probability measure on $\mathfrak{B}(\mathbb{R})$? I've been working through online lectures on probability theory and I've hit a conceptual snag. (There's no way to contact the presenter since these are several years old.)
Hed efined an algebra on (0,1] and a pseudo probability on it that returned 1 for the sample space. He did not prove additivity but gave a reference and then invoked Caratheorary's theorem to get a probability measure on the sigma algebra generated by the algebra. The latter he showed was the Borel sigma algebra.
The pseudo probability he chose was $\lambda((a,b)) = b-a$ for $b >a$. Thus he established $((0,1], \mathfrak{B}((0,1]), \lambda)$ as a probability space.
Now while it seems clear that the Borel sigma algebra can be extended to $\mathbb{R}$, the triple $(\mathbb{R}, \mathfrak{B}(\mathbb{R}), \lambda)$ is not a probability space unless it is restricted to finite subsets of $\mathbb{R}$. Does this mean there is $\textbf{no}$ probability measure possible on the Borel sigma algebra on $\mathbb{R}$, or just no uniform measure?
And follow up, if I may. I see that $\lambda$ is infinite on all of $\mathbb{R}$ and so cannot be a probability measure. But what if I took the same approach as above with an algebra on subsets of $\mathbb{R}$ and$\lambda$ as the pseudo measure---why would Caratheodory's theorem not result in a uniform probability measure on $\mathfrak{B}(\mathbb{R})$? (A remarkable theorem!)
 A: Here’s a more complete answer than the one I gave in the comment.
There is a uniform measure on $\frak{B}(\mathbb{R})$. As you noted, we can use Carotheodory extension to get a measure $m$ with the property that $m((a,b)) = b - a$ for any open interval $(a,b) \subseteq \mathbb{R}$. You also correctly noted that this is not a probability measure, as $m(\mathbb{R}) = \infty$. This measure is commonly called the “Lebesgue measure” on $\mathbb{R}$.
It is worth noting that the Lebesgue measure can be extended to a larger $\sigma$-algebra than $\frak{B}(\mathbb{R})$. This larger $\sigma$-algebra is (unsurprisingly) called the “Lebesgue $\sigma$-algebra”. However for probability, we usually only work with the Borel sets.
On the other hand, there are loads of different probability measures that can be defined on $\frak{B}(\mathbb{R})$. The so-called “laws” of random variables give us a rich class of examples. Recall that given a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a real-valued random variable is a function $X:\Omega \rightarrow \mathbb{R}$ such that for any Borel set $B \in \frak{B}(\mathbb{R})$, we have that the preimage $X^{-1}(B) \in \mathcal{F}$.
So let $X:\Omega \rightarrow \mathbb{R}$ be a random variable. There is an associated probability measure on $\mathbb{R}$ called the “law” of $X$. This measure is commonly denoted $\mathcal{L}_X$. This measure is achieved by “pushing $\mathbb{P}$ forward” by $X$. That is, for any Borel set $B \in \frak{B}(\mathbb{R})$, we set:
$$L_X(B) := \mathbb{P}(\omega \in \Omega : X(\omega) \in B)$$
That is, the measure of a set is the probability that $X$ “ends up” in the set.
