  Algorithmic randomness theory starts with a notion of an individual random
object. To be reasonable, this notion should have some natural properties; in
particular, an object should be random with respect to image distribution if
and only if it has a random preimage. This result (for computable distributions
and mappings, and Martin-L\"of randomness) was known for a long time
(folklore); in this paper we prove its natural generalization for layerwise
computable mappings, and discuss the related quantitative results.
