Numberings and randomness

We prove various results on effective numberings and Friedberg numberings of families related to algorithmic randomness. The family of all Martin-L"of random left-computably enumerable reals has a Friedberg numbering, as does the family of all ${\mathrm{\Pi }}_1^0$ classes of positive measure. On the other hand, the ${\mathrm{\Pi }}_1^0$ classes contained in the Martin-L"of random reals do not even have an effective numbering, nor do the left-c.e. reals satisfying a fixed randomness constant. For ${\mathrm{\Pi }}_1^0$ classes contained in the class of reals satisfying a fixed randomness constant, we prove that at least an effective numbering exists.