| context | $ \langle X,\Sigma_X\rangle\in \mathrm{MeasurableSpace}(X) $ |
| context | $ \langle Y,\Sigma_Y\rangle\in \mathrm{MeasurableSpace}(Y) $ |
| postulate | $ f\in \mathrm{Measurable}(X,Y) $ |
| context | $ f:X\to Y $ |
| $y\in \Sigma_Y$ |
| postulate | $ f^{-1}(y)\in\Sigma_X $ |
This is very similar to the definition of continuous function.
People write $f:\langle X,\Sigma_X\rangle\to\langle Y,\Sigma_Y\rangle$ to point out the function is measurable, although I'd say that's abuse of language.
Wikipedia: Sigma-algebra