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An automorphism of a measure space, which describes [[Bernoulli trials|Bernoulli trials]] and their generalization — a sequence of independent trials with the same result and with the same probability distribution.

Let  $  A $
be the collection of all possible outcomes of a trial, and let the probability of the event  $  B \subset  A $
be given by the measure  $  \nu $; 
for a countable set  $  A $, 
denote its elements by  $  a _ {i} $
and their probabilities by  $  p _ {i} = \nu (a _ {i} ) $. 
The phase space of a Bernoulli automorphism is the direct product of a countable number of copies of the set  $  A $, 
i.e. the points of the phase space are infinite sequences  $  b = \{ b _ {k} \} $, 
where  $  k $
runs through the set of integers and each  $  b _ {k} \in A $. 
The transformation  $  T $
consists in shifting all members of each sequence one place to the left:  $  T \{ b _ {k} \} = \{ b _ {k+1 }  \} $. 
The measure  $  \mu $
is defined as the direct product of a countable number of measures  $  \nu $; 
thus if  $  A $
is countable, then

$$ 
\mu \{ {b } : {b _ {i _ {1}  } =
a _ {j _ {1}  } \dots b _ {i _ {k}  } =
a _ {j _ {k}  } } \}
 = \ 
p _ {j _ {1}  } \dots p _ {j _ {k}  } .
$$

In this case, the [[Entropy|entropy]] of the Bernoulli automorphism is  $  - \sum p _ {i}   \mathop{\rm log}  p _ {i} $.

In ergodic theory, Bernoulli automorphisms (or, more exactly, the cascades generated by iteration of them) play the role of a standard example of a dynamical system, the behaviour of which displays statistical features. A Bernoulli automorphism is a  $  K $-
automorphism but there exist  $  K $-
automorphisms which are metrically non-isomorphic to a Bernoulli automorphism, even though many  $  K $-
automorphisms are metrically isomorphic to a Bernoulli automorphism. Two Bernoulli automorphisms are metrically isomorphic if and only if they have the same entropy . A Bernoulli automorphism is a quotient automorphism of any ergodic automorphism of a Lebesgue space with a larger entropy [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  D. Ornstein,   "Bernoulli shifts with the same entropy are isomorphic"  ''Adv. Math.'' , '''4'''  (1970)  pp. 337–352</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  D.Ornstein,   "A Kolmogorov automorphism that is not a Bernoulli shift"  ''Matematika'' , '''15''' :  1  (1971)  pp. 131–150  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ya.G. Sinai,   "On weak isomorphism of transformations with invariant measure"  ''Mat. Sb.'' , '''63 (105)''' :  1  (1964)  pp. 23–42  (In Russian)</TD></TR></table>

====Comments====
A metric isomorphism between two Bernoulli automorphisms can be given by means of a finitary code ([[#References|[a1]]], see also [[#References|[a4]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Keane,   M. Smorodinsky,   "Bernoulli schemes with the same entropy are finitarily isomorphic"  ''Ann. of Math.'' , '''109'''  (1979)  pp. 397–406</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Smorodinsky,   "Ergodic theory, entropy" , Springer  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Ornstein,   "Ergodic theory, randomness, and dynamical systems" , Yale Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I.P. [I.P. Kornfel'd] Cornfel'd,   S.V. Fomin,   Ya.G. Sinai,   "Ergodic theory" , Springer  (1982)  (Translated from Russian)</TD></TR></table>