  We study functions on the infinite-dimensional Hamming cube
$\{-1,1\}^\infty$, in particular Boolean functions into $\{-1,1\}$,
generalising results on analysis of Boolean functions on $\{-1,1\}^n$ for
$n\in\mathbb{N}$. The notion of noise sensitivity, first studied in
arXiv:math/9811157 , is extended to this setting, and basic Fourier formulas
are established. We also prove hypercontractivity estimates for these
functions, and give a version of the Kahn-Kalai-Linial theorem giving a bound
relating the total influence to the maximal influence.
  Particular attention is paid to so-called finitary functions, which are
functions for which there exists an algorithm that almost surely queries only
finitely many bits. Two versions of the Benjamini-Kalai-Schramm theorem
characterizing noise sensitivity in terms of the sum of squared influences are
given, under additional moment hypotheses on the amount of bits looked at by an
algorithm. A version of the Kahn-Kalai-Linial theorem giving that the maximal
influence is of order $\frac{\log(n)}{n}$ is also given, replacing $n$ with the
expected number of bits looked at by an algorithm.
  Finally, we show that the result in arXiv:math/0504586 that revealments going
to zero implies noise sensitivity also holds for finitary functions, and apply
this to show noise sensitivity of a version of the voter model on sufficiently
sparse graphs.
