We study algorithmic randomness with respect to c.e. semimeasures, which naturally arise as pushforward measures of partial computable mappings and therefore play a crucial role in algorithmic randomness. We consider four distinct randomness notions: three based on complexity and one based on tests. We systematically clarify their inclusion relationships. Our main contribution is to construct concrete examples that separate these notions. Furthermore, we investigate how they interact with the classical randomness preservation and no-randomness-from-nothing theorems, identifying precise conditions under which they continue to hold.

We propose studying uniform Kurtz randomness, which is the uniform relativization of Kurtz randomness. This notion has more natural properties than the usual relativization. For instance, van Lambalgen's theorem holds for uniform Kurtz randomness but not for (the usual relativization of) Kurtz randomness. Another advantage is that lowness for uniform Kurtz randomness has many characterizations, such as those via complexity, martingales, Kurtz tt-traceability and Kurtz dimensional measure.