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''measurable partition, of a [[Measure space|measure space]]  $  ( M, \mu ) $''

A partition (cf. [[Decomposition|Decomposition]])  $  \xi $
of the space into disjoint subsets (called the elements of the partition) that can be obtained as the partition into level sets of some [[Measurable function|measurable function]] (with numerical values) on  $  M $. 
This definition can be restated in terms of  "intrinsic"  properties of the partition (see [[#References|[1]]]). In accordance with the general tendency in questions of measure theory to ignore sets of measure zero, a measurable partition is often understood to be a partition measurable modulo 0, that is, equivalent modulo 0 to some measurable partition (two partitions  $  \xi $
and  $  \eta $
of a measure space  $  M $
are equivalent modulo 0 if there exists a set  $  N $
of measure 0 such that the partitions of  $  M \setminus  N $
consisting of the intersections with  $  M \setminus  N $
of the elements of  $  \xi $
and  $  \eta $
coincide).

Although the definition given makes sense for any measure space, in fact measurable partitions are almost always considered for Lebesgue spaces (cf. [[Lebesgue space|Lebesgue space]]) (and sometimes for spaces having the properties of the latter to some extent, for example, for spaces with perfect measures; see [[#References|[2]]] and [[#References|[3]]] and [[Perfect measure|Perfect measure]]), since in these spaces measurable partitions have a number of  "good"  properties. Thus, in this case there exists a system of conditional measures (or, as used to be said in earlier times [[#References|[1]]], a canonical system of measures) belonging to measurable partitions. This is a system of measures  $  \mu ( \cdot \mid  C) $
on the elements  $  C $
of the partition  $  \xi $
which enables one to consider integration with respect to  $  \mu $
as repeated integration: first, integration is performed over the  $  C $
and with respect to appropriate  $  \mu ( \cdot \mid  C) $, 
and then it is necessary to integrate the result, which can be looked upon as a function on the quotient space  $  M/ \xi $, 
with respect to the natural measure  $  \mu _  \xi  $
on the latter space (by definition,  $  M/ \xi $
has the elements of  $  \xi $
as points, while its measurable subsets are those with measurable pre-images under the natural projection  $  \pi :  M \rightarrow M/ \xi $; 
the measure is defined to be  $  \mu _  \xi  ( A) = \mu ( \pi  ^ {-} 1 ( A)) $). 
Interpreting  $  ( M, \mu ) $
as a space of elementary events in probability theory, one can say that the system of conditional measures is an  "improvement"  of the conditional probability, closely connected with the specific Lebesgue space; for arbitrary spaces of elementary events, the conditional probability cannot, in general, be interpreted using some set of measures on the elements of some partition.

Non-measurable (and non-measurable modulo 0) partitions are by no means always  "pathological"  objects, like non-measurable sets or functions. For example, the partition of the phase space of an ergodic dynamical system into its trajectories can have a completely  "classical"  origin; it is simply that its properties are different from those of a measurable partition.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Rokhlin,   "On mean notions of measure theory"  ''Mat. Sb.'' , '''25''' :  1  (1949)  pp. 107–150  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Gnedenko,   A.N. Kolmogorov,   "Limit distributions for sums of independent random variables" , Addison-Wesley  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Sazonov,   "On perfect measures"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''26''' :  3  (1962)  pp. 391–414  (In Russian)</TD></TR></table>

====Comments====
A useful result in ergodic theory concerning Lebesgue spaces and measurable partitions is the Rokhlin–Halmos theorem. In its strong form it states: Let  $  \pi = ( A _ {1} \dots A _ {r} ) $
be a finite measurable partition of a Lebesgue space  $  ( M, {\mathcal A} , \mu ) $
and let  $  T $
be an [[Automorphism|automorphism]] of this space. Then, for any  $  \epsilon > 0 $
and any  $  n \in \mathbf N $
there exists an  $  E \in {\mathcal A} $
such that  $  E, TE \dots T  ^ {n-} 1 E $
are pairwise disjoint,  $  \mu ( \cup _ {i=} 0  ^ {n-} 1 T  ^ {i} E) > 1- \epsilon $
and  $  \mu ( E \cap A _ {j} ) = \mu ( E) \mu ( A _ {j} ) $, 
$  j = 1 \dots r $. 
(The weak form results by taking the trivial partition  $  ( M) $.)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.P. [I.P. Kornfel'd] Cornfel'd,   S.V. Fomin,   Ya.G. Sinai,   "Ergodic theory" , Springer  (1982)  pp. Chapt. 10, §5  (Translated from Russian)</TD></TR></table>