Highness properties close to PA completeness

Suppose we are given a computably enumerable object. We interested in the strength of oracles that can compute an object approximates this c.e. It turns out many cases arising from algorithmic randomness or computable analysis, resulting highness property is either close to, equivalent to being PA complete. examine, for example, majorizing martingale by oracle-computable martingale, computing lower bounds two variants Kolmogorov complexity, and subtree positive measure with no dead-ends $$\prod _1^0$$ ∏ 1 0 tree measure. separate completeness latter property, called continuous covering property. also corresponding principles reverse mathematics.