Q: Please let me know if the following sketch of the build-up of the Steinhaus probability space $(\Omega,{\cal A},P) = ((0,1), {\cal B}((0,1)), \lambda\vert_{[0,1]})$ is correct, as far as it goes:
Let the sample space $\Omega=(0,1)$, the real open interval from 0 to 1, be an open set in $2^\Re$.
Let $d(a,b)=|b-a|, a,b\in \Re$ be a metric.
Then $(\Omega,d)$ is a metric space, because $(0,1)$ is a nonempty set and $d$ is a metric on $(0,1)$.
Let ${\cal O}(\Omega)$ be the collection of all open sets $F \subseteq \Omega$.
Let $\Sigma({\cal O}(\Omega))$ be the generated $\sigma$-algebra of ${\cal O}(\Omega)$.
Then $(\Omega, \Sigma({\cal O}(\Omega)))$ is a measurable space, because $(0,1)$ is a set and $\Sigma({\cal O}(0,1))$ is a $\sigma$-algebra on that set.
Then ${\cal B}((0,1)) = {\cal B}(\Omega) = \Sigma({\cal O}(\Omega))$ is the Borel $\sigma$-algebra of $(0,1)$, because $((0,1),d)$ is a metric space and ${\cal O}((0,1))$ is the set of all open sets of $(0,1)$.
Let $\lambda: {\cal L}(\Re) \to [0,\infty)$ be the Lebesgue measure on $\Re$.
Let $\lambda\vert_{(0,1)}$ be the retriction of $\lambda$ to $\Sigma({\cal O}((0,1)))$.
Then $\lambda\vert_{(0,1)}$ is a probability measure, because $((0,1), \Sigma({\cal O}((0,1))))$ is a measurable space and $\lambda\vert_{(0,1)}((0,1)) = 1$.
Then $((0,1), {\cal B}((0,1)), \lambda\vert_{[0,1]})$ is a probability space, since $(0,1)$ is a set, ${\cal B}((0,1))$ is a $\sigma$-algebra on that set, and $\lambda\vert_{[0,1]}$ is a probability measure on that set.
Note: This will seem very elementary to anybody that knows this topic. I am new to the topic and trying to check my understanding, with a view towards implementing these definitions and constructions as Python classes and symbolic computations in SymPy. So I'm trying to make each step in the build-up be something well-defined, explicit and mechanically checkable to some extent.