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''of a set in a metric space''

The logarithm to the base 2 of the smallest number of points in an  $  \epsilon $-
net for this set. In other words, the  $  \epsilon $-
entropy  $  {\mathcal H} _  \epsilon  ( C , X ) $
of a set  $  C $
lying in a metric space  $  ( X , \rho ) $
is

$$ 
{\mathcal H} _  \epsilon  ( C , X )  = \ 
 \mathop{\rm log}  _ {2} N _  \epsilon  ( C , X ) ,
$$

where

$$ 
N _  \epsilon  ( C , X )  =  \inf \left \{ {n } : {
\exists x _ {1} \dots x _ {n} , x _ {i} \in X :\ 
C \subset  \cup _ { i= } 1 ^ { n }  B ( x _ {i} , \epsilon ) } \right \}
$$

and  $  B ( \zeta , \alpha ) = \{ {x \in X } : {\rho ( x , \zeta ) \leq  \alpha } \} $
is a ball of radius  $  \alpha $
with centre at  $  \zeta $. 
The definition of  $  {\mathcal H} _  \epsilon  ( C , X ) $
was given by A.N. Kolmogorov in [[#References|[1]]], motivated by ideas and definitions of information theory (cf. [[Information theory|Information theory]]).  $  {\mathcal H} _  \epsilon  ( C , X ) $
is also called the relative  $  \epsilon $-
entropy; it depends on the space  $  X $
in which the set  $  C $
is situated, that is, on the metric extension of  $  C $. 
The quantity

$$ 
{\mathcal H} _  \epsilon  ( C)  =  \inf  {\mathcal H} _  \epsilon  ( C , X ) ,
$$

where the infimum is taken over all metric extensions  $  X $
of  $  C $, 
is called the absolute  $  \epsilon $-
entropy of  $  C $. 
It may also be defined directly (Kolmogorov, [[#References|[1]]]): for a metric space  $  C $, 
$  {\mathcal H} _  \epsilon  ( C) $
is the logarithm to the base 2 of the cardinality  $  N _  \epsilon  ( C) $
of the most economic (by the number of sets)  $  2 \epsilon $-
covering of  $  C $. 
Here a system of sets  $  \{ C _ {i} \} $
is called a  $  2 \epsilon $-
covering of  $  C $
if  $  C _ {i} \subset  C $, 
$  \cup _ {i=} 1  ^ {n} C _ {i} = C $
and the diameter of each set does not exceed  $  2 \epsilon $. 
It has been shown [[#References|[2]]] that the absolute  $  \epsilon $-
entropy is the minimum relative  $  \epsilon $-
entropy. The quantities  $  N _  \epsilon  ( C) $
and  $  N _  \epsilon  ( C , X ) $
are inverse to the widths (cf. [[Width|Width]])  $  \epsilon  ^ {N} ( C) $
and  $  \epsilon _ {N} ( C , X ) $. 
This characterizes the fact that it is possible to recover the elements of  $  C $
from tables of  $  N $
elements and to approximate by  $  N $-
point sets.

The investigation of the asymptotic behaviour of the  $  \epsilon $-
entropy of various function classes is a special branch of approximation theory.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  A.N. Kolmogorov,   "On certain asymptotic characteristics of completely bounded metric spaces"  ''Dokl. Akad. Nauk SSSR'' , '''108''' :  3  (1956)  pp. 385–388  (In Russian)</td></tr><tr><td valign="top">[2]</td> <td valign="top">  A.G. Vitushkin,   "Theory of transmission and processing of information" , Pergamon  (1961)  (Translated from Russian)</td></tr><tr><td valign="top">[3]</td> <td valign="top">  A.N. Kolmogorov,   V.M. Tikhomirov,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035000/e03500046.png"/>-entropy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035000/e03500047.png"/>-capacity of sets in functional spaces"  ''Amer. Math. Soc. Transl. Ser. 2'' , '''17'''  (1961)  pp. 277–364  ''Uspekhi Mat. Nauk'' , '''14''' :  2  (1959)  pp. 3–86</td></tr></table>

====Comments====
Kolmogorov's original definition was formulated to investigate the question of whether functions of  $  n \geq  3 $
variables are representable as compositions of functions of  $  r &lt; n $
variables. This has a great deal to do with Hilbert's 13th problem, cf. [[#References|[2]]] and [[#References|[a1]]].

The notion of  $  \epsilon $-
entropy has also proved useful in probability theory, cf. [[#References|[a2]]]. It is also called metric entropy.

The  $  \epsilon $-
entropy  $  {\mathcal H} _  \epsilon  ( C , X) $
is defined by considering coverings of  $  C $
by  $  \epsilon $-
balls. An  $  \epsilon $-
packing  $  {\mathcal P} = \{ B ( y _ {i} , \epsilon ) \} $
of  $  C $
is a collection of  $  \epsilon $-
balls such that  $  B( y _ {i} , \epsilon ) \cap B ( y _ {j} , \epsilon ) = \emptyset $
if  $  i \neq j $
and  $  B ( x _ {i} , \epsilon ) \subset  C $. 
Let  $  M ( C , \epsilon ) $
be the maximum cardinality of a packing in  $  C $. 
Then  $   \mathop{\rm log}  M( C , \epsilon ) $
is called the  $  \epsilon $-
capacity of  $  C $
in  $  X $.

There are a number of additional (but not unrelated) notions of  $  \epsilon $-
entropy for random variables and random processes. They are defined as follows. Let  $  \xi $
be a random variable with values in a metric space  $  ( X , \rho ) $. 
Let  $  W _  \epsilon  $
be the set of all random variables  $  \xi  ^  \prime  $
with values in  $  X $
such that  $  {\mathsf E} ( \rho  ^ {2} ( \xi , \xi  ^  \prime  ) ) \leq  \epsilon  ^ {2} $, 
where  $  {\mathsf E} $
denotes mathematical expectation. Then the  $  \epsilon $-
entropy of  $  \xi $
is ([[#References|[a3]]])

$$ 
{\mathcal H} _  \epsilon   ^  \prime  ( \xi )  = \ 
\inf \{ {I ( \xi , \xi  ^  \prime  ) } : {
\xi  ^  \prime  \in W _  \epsilon  } \}
,
$$

where  $  I ( \xi , \xi  ^  \prime  ) $
is the usual information function for a pair of random variables (cf. [[Information|Information]]). This idea can be easily extended to random processes  $  \{ \xi ( t) \} _ {t \in [ a, b ] }  $
by regarding such a process as a random variable with values in  $  L _ {2} ([ a , b ]) $, 
say. [[#References|[a4]]] contains some asymptotic estimates of  $  {\mathcal H} _  \epsilon   ^  \prime  ( \xi ) $.

Now let  $  ( X , \rho , \mu ) $
be a probabilistic metric space, i.e.  $  ( X, \rho ) $
is a metric space and  $  ( X , \mu ) $
is a probability space. For  $  \epsilon &gt; 0 $, 
let  $  {\mathcal A} _  \epsilon  $
be the set of all partitions of  $  X $(
cf. [[Decomposition|Decomposition]]) into measurable sets of diameter  $  \leq  \epsilon $. 
Then ([[#References|[a5]]]) one defines

$$ 
{\mathcal H} _  \epsilon   ^ {\prime\prime} ( X)  = \ 
\inf \{ {H ( {\mathcal U} ) } : {
{\mathcal U} \in {\mathcal A} _  \epsilon  } \}
,
$$

where  $  H ( {\mathcal U} ) $
denotes the ordinary entropy of the partition  $  {\mathcal U} $. 
Define

$$ 
I _  \epsilon  ( X)  =  \lim\limits _ {n \rightarrow \infty } \ 

\frac{1}{n}
 {\mathcal H} _  \epsilon   ^ {\prime\prime} ( X  ^ {n} ) ,
$$

where  $  X  ^ {n} = X \times \dots \times X $(
$  n $
factors) with the product measure and distance. (This limit exists.) For a measure  $  \pi $
on  $  X \times X $
such that  $  \pi ( A \times X ) = \pi ( X \times A) = \mu ( A) $
for all measurable  $  A \subset  X $, 
let

$$ 
I ( \rho )  =  
\frac{d \rho }{d ( \mu \times \mu ) }

$$

be the Radon–Nikodým derivative. Let  $  R _  \epsilon  ( X) $
be the set of all measures  $  \pi $
on the measurable space  $  X \times X $
such that  $  \pi ( A \times X ) = \pi ( X \times A) = \mu ( A) $
for  $  A $
measurable and

$$ 
\pi \{ {( x, y) } : {\rho ( x, y) \leq  \epsilon /2 } \}
 =  1.
$$

Then

$$ 
I _  \epsilon   =  \inf _ {\rho \in R _  \epsilon  ( X) } \ 
I ( \rho ) ,
$$

which is a noisy channel coding theorem; cf. [[#References|[a6]]]. There are also estimates both ways relating  $  {\mathcal H} _  \epsilon   ^ {\prime\prime} $
and  $  I _  \epsilon  ( X) $; 
cf. [[#References|[a6]]]. Cf. also [[#References|[a7]]] for (more) relations between  $  {\mathcal H} _  \epsilon   ^ {\prime\prime} $
and (extended notions of) Shannon entropy.

Define the absolute  $  \epsilon $-
entropy  $  {\mathcal H} _  \epsilon  ( C) $(
as in the main article above) by means of  $  \epsilon $-
partitions of  $  ( X , \rho ) $
by means of Borel sets. Then

$$ 
{\mathcal H} _  \epsilon   ^ {\prime\prime}  \leq   {\mathcal H} _ {\epsilon /2 }  ,
$$

but it need not be true that the supremum of the  $  {\mathcal H} _  \epsilon   ^ {\prime\prime} $
for varying  $  \mu $
on  $  ( X, \rho ) $
is equal to  $  {\mathcal H} _ {\epsilon /2 }  $, 
cf. [[#References|[a5]]].

Now let again  $  \xi ( t) $, 
$  t \in [ 0, 1] $, 
be a stochastic Gaussian process (real-valued), let  $  K ( \cdot , \cdot ) $
be its covariance function and let  $  T $
be the integral operator with kernel  $  K( \cdot , \cdot ) $, 
i.e.

$$ 
( Tf  ) ( t)  =  \int\limits _ { 0 } ^ { 1 }  K( s, t) f( s)  ds .
$$

Let  $  \lambda _ {1} \geq  \lambda _ {2} \geq  \dots \geq  0 $
be the eigen values of  $  T $. 
For  $  \epsilon \in [ 0, ( \sum _ {i=} 1  ^  \infty  \lambda _ {i} )  ^ {1/2} ] $
define a function  $  f ( \epsilon ) $
by the equality

$$ 
\epsilon  ^ {2}  = \ 
\sum _ { i= } 1 ^  \infty   \min \{ \lambda _ {i} , f ( \epsilon ) \} .
$$

Then Pinsker's theorem says that

$$ 
{\mathcal H} _  \epsilon   ^  \prime  ( \xi )  =  
\frac{1}{2}

\sum _ { i= } 1 ^  \infty    \mathop{\rm log}  \max \ 
\left \{ 
\frac{\lambda _ {i} }{f ( \epsilon ) }
 , 1 \right \}
$$

(where log stands for the logarithm to the base 2). One also has ([[#References|[a8]]]): For all  $  a &gt; 0 $,

$$ 
\lim\limits _ {k \rightarrow \infty }  
\frac{S( T  ^ {k} , a f ( \epsilon )  ^ {k} ) }{k}

 =  2 {\mathcal H} _  \epsilon   ^  \prime  ( \xi ) ,
$$

where for a suitable operator on a Hilbert space  $  T :  H \rightarrow H $
the entropy  $  S( T, \alpha ) $
is defined as the metric entropy of the set  $  \{ {Tx } : {\| x \| \leq  1 } \} \subset  H $:

$$ 
S ( T , \alpha )  =  {\mathcal H} _  \alpha  ( T( B( 0, 1)) , H ).
$$

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  G.G. Lorentz,   "The 13-th problem of Hilbert"  F.E. Browder (ed.) , ''Mathematical developments arising from Hilbert problems'' , ''Proc. Symp. Pure Math.'' , '''28''' , Amer. Math. Soc.  (1976)  pp. 419–430</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.M. Dudley,   "Metric entropy and the central limit theorem in $C ( S )$"  ''Ann. Inst. Fourier (Grenoble)'' , '''24'''  (1974)  pp. 49–60</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A.N. Kolmogorov,   "Theory of transmission of information"  ''Amer. Math. Soc. Transl. Ser. 2'' , '''33'''  (1963)  pp. 291–321  ''Acad. R. P. Romine An. Romino-Sov.'' , '''13''' :  1  (1959)  pp. 5–33</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Biria,   M. Zakai,   J. Ziv,   "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035000/e035000141.png"/>-entropy and the rate distortion function of certain non-Gaussian processes"  ''IEEE Trans. Inform. Theory'' , '''20'''  (1974)  pp. 517–524</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  E.C. Posner,   E.R. Rodenich,   H. Rumsey,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035000/e035000142.png"/>-Entropy of stochastic processes"  ''Ann. Math. Stat.'' , '''38'''  (1967)  pp. 1000–1020</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E.C. Posner,   E.R. Rodenich,   "Epsilon entropy and data compression"  ''Ann. Math. Stat.'' , '''42'''  (1971)  pp. 2079–2125</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  M. Katětov,   "On extended Shannon entropies and the epsilon entropy"  ''Comm. Math. Univ. Carolinae'' , '''27'''  (1986)  pp. 519–534</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  S. Akashi,   "On operator theoretical characterization of epsilon-entropy in Gaussian processes"  ''Kodai Math. J.'' , '''9'''  (1986)  pp. 58–67</td></tr></table>