  The notion of a normal bit sequence was introduced by Borel in 1909; it was
the first definition of an individual random object. Normality is a weak notion
of randomness requiring only that all $2^n$ factors (substrings) of arbitrary
length~$n$ appear with the same limit frequency $2^{-n}$. Later many stronger
definitions of randomness were introduced, and in this context normality found
its place as ``randomness against a finite-memory adversary''. A quantitative
measure of finite-state compressibility was also introduced (the finite-state
dimension) and normality means that the finite state dimension is maximal
(equals~$1$).
  Recently Nandakumar, Pulari and S (2023) introduced the notion of relative
finite-state dimension for a binary sequence with respect to some other binary
sequence (treated as an oracle), and the corresponding notion of conditional
(relative) normality. (Different notions of conditional randomness were
considered before, but not for the finite memory case.) They establish
equivalence between the block frequency and the gambling approaches to
conditional normality and finite-state dimensions.
  In this note we revisit their definitions and explain how this equivalence
can be obtained easily by generalizing known characterizations of
(unconditional) normality and dimension in terms of compressibility
(finite-state complexity), superadditive complexity measures and gambling
(finite-state gales), thus also answering some questions left open in the
above-mentioned paper.
