A Sigma-algebra, $\mathcal{B}$, on a set, $\Omega$, defined as:

$\mathcal{B} = \sigma(\tau)$

i.e. the sigma-algebra generated by $\tau$, which is the set:{all the open sets of $\Omega$}, i.e. the topology on $\Omega$. It is the smallest sigma-algebra that contains $\tau$. See here

A Borel measure, is just a Measure on a Borel $\sigma$-algebra. Specifying such a measure is simplified by the Caratheodory extension theorem, that says that to