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A quantitative expression for the error of an approximation. When the discussion is about the approximation of a function  $  f $
by a function  $  \phi $, 
the measure of approximation  $  \mu (f, \phi ) $
is usually defined by the metric in a function space containing both  $  f $
and  $  \phi $. 
For example, if  $  f $
and  $  \phi $
are continuous functions on a segment  $  [a, b] $, 
the uniform metric of  $  C [a, b] $
is commonly used, i.e. one puts

$$ 
\mu (f, \phi )  = \ 
\max _ {a \leq  t \leq  b } \ 
| f (t) - \phi (t) | .
$$

If continuity of the approximated function is not guaranteed or if the conditions of the problem imply that it is important that  $  f $
and  $  \phi $
are close on  $  [a, b] $
in an average sense, the integral metric of a space  $  L _ {p} [a, b] $
may be used, putting

$$ 
\mu (f, \phi )  = \ 
\int\limits _ { a } ^ { b } 
q (t) | f (t) -
\phi (t) |  ^ {p} \ 
dt,\  p > 0,
$$

where  $  q (t) $
is a weight function. The case  $  p = 2 $
is most often used and is most convenient from a practical point of view (cf. [[Mean-square approximation of a function|Mean-square approximation of a function]]).

The measure of approximation may take into account only values of  $  f $
and  $  \phi $
in discrete points  $  t _ {k} $, 
$  k = 1 \dots n $, 
of  $  [a, b] $, 
e.g.

$$ 
\mu (f, \phi )  = \ 
\max _ {1 \leq  k \leq  n } \ 
| f (t _ {k} ) - \phi (t _ {k} ) | ,
$$

$$ 
\mu (f, \phi )  =  \sum _ {k = 1 } ^ { n }  q _ {k} | f (t _ {k} ) - \phi (t _ {k} ) |  ^ {p} ,
$$

where  $  q _ {k} $
are certain positive coefficients.

One defines in an analogous way the measure of approximation of functions in two or more variables.

The measure of approximation of a function  $  f $
by a family  $  F $
of functions is usually taken to be the [[Best approximation|best approximation]]:

$$ 
E (f, F)  = \ 
\mu (f, F)  = \ 
\inf _ {\phi \in F } \ 
\mu (f, \phi ).
$$

The quantity

$$ 
E ( \mathfrak M , F)  = \ 
\mu ( \mathfrak M , F)  = \ 
\sup _ {f \in \mathfrak M } \ 
\inf _ {\phi \in F } \ 
\mu (f, \phi )
$$

is usually taken as the measure of approximation of a class  $  \mathfrak M $
of functions  $  f $
by functions  $  \phi $
from a certain fixed set  $  F $. 
It characterizes the maximal deviation of functions in  $  \mathfrak M $
from functions in  $  F $
that are closest to them.

In general, when approximation in an arbitrary metric space  $  X $
is considered, the measure of approximation  $  \mu (x, u) $
of an element  $  x $
by an element  $  u $(
a set  $  F $) 
is the distance  $  \rho (x, u) $(
or  $  \rho (x, F) $) 
between  $  x $
and  $  u $(
or  $  F $) 
in the metric of  $  X $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.L. Goncharov,   "The theory of interpolation and approximation of functions" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R. Rice,   "The approximation of functions" , '''1–2''' , Addison-Wesley  (1964–1968)</TD></TR></table>

====Comments====
The measure of approximation is also called the error measure.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.G. Lorentz,   "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Pinkus,   "$n$-widths in approximation theory" , Springer  (1985)  (Translated from Russian)</TD></TR>
</table>