First of all: thanks to Rupert for putting up this blog. It looks great! My first post will be a question which I stumbled upon while working on effective Brownian motion with Kelty Allen:
Let $X$ be an ML random real and $Y$ some real. Is there always a sequence $R$ of rationals converging to $Y$ such that $X$ is random relative to $R$?
Of course the only interesting case is when $X$ is not ML-random relative to $Y$. Note that the answer is no if we further ask $R$ to be nondecreasing (take $X=\Omega$ and $Y=1-\Omega$: any nondecreasing sequence converging to $1-\Omega$ computes $1-\Omega$ so it derandomizes $X$)