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100 XIII INTEGRATION

This definition agrees with the preceding one when X is compact. It
expresses that the restriction of ju to Jf(X; K) is continuous with respect to
the topology induced by that of ^(X). We remark that tf (X; K) is closed
in ^c(X), and therefore a Banach space (3.14.5).

In general, a measure is not necessarily continuous on «3T_C(X) with respect
to the topology induced by that of ^(X) (i.e., the topology defined by the
norm ||/||). We shall examine this question later (13.20).

Examples of Measures

(13.1.3) Let X be a locally compact space, and let xeX. The mapping
/i->/(jc) of Jf (X) into C is a measure, for it is linear and we have \f(x)\ ^ ||/||
for each compact subset K of X such that/e Jf (X; K). This measure is called
the Dirac measure at the point x, or the measure defined by the unit mass at
the point jc, and is denoted by &_x.

More generally, let (a_n) be a sequence of distinct points in X, and (/„) a
sequence of complex numbers, such that for each compact K c X the sub-
sequence formed by the t_nfor which a_n e K is absolutely summable (5.3). Let
^CK — Z l*nl- Then, for each function /e «^T(X; K), the series ]T t_nf(a_n) is

absolutely convergent, because the only nonzero terms are those for which
a_n e K, and we have

£ \tJ(a_n)\ S II/H • E t_n = c_K-\\f\\.

a_n e K a_n £ K

This also shows that/i->£ ^/fe) is a measure on X. This measure is said to

be defined by the masses t_n at the points a_n for all n (cf. (13.18.8)).

(13.1.4) Let/e JT_CW- F°^r ^each interval [a, b] containing the support of/,
the value of the integral /(/) dt (8.7) is the same, and we denote it by

/»+ oo /*+ oo

f(f) dt. The mapping/r-^ /(/) dt is a linear form on JT_COR)> ^an(i it is

J ~~ oo J — oo

a measure because, for each compact interval K = [a, b~] in R and each func-
tion/e jf(R; K) we have

+ 00

_J(t)i

by the mean value theorem (8.7.7). This measure is called Lebesgue measure
on the real line R.

(13.1.5) Let fj. be a measure on X and let g 6 #_c(^x)- Then for each function
/e X(X) it is clear that #/e jf (X), and the mapping/H-»JU(#/) is therefore



	100 XIII INTEGRATION This definition agrees with the preceding one when X is compact. It expresses that the restriction of ju to Jf(X; K) is continuous with respect to the topology induced by that of ^(X). We remark that tf (X; K) is closed in ^c(X), and therefore a Banach space (3.14.5). In general, a measure is not necessarily continuous on «3T_C(X) with respect to the topology induced by that of ^(X) (i.e., the topology defined by the norm \|\|/\|\|). We shall examine this question later (13.20). Examples of Measures (13.1.3) Let X be a locally compact space, and let xeX. The mapping /i->/(jc) of Jf (X) into C is a measure, for it is linear and we have \f(x)\ ^ \|\|/\|\| for each compact subset K of X such that/e Jf (X; K). This measure is called the Dirac measure at the point x, or the measure defined by the unit mass at the point jc, and is denoted by &_x. More generally, let (a_n) be a sequence of distinct points in X, and (/„) a sequence of complex numbers, such that for each compact K c X the sub- sequence formed by the t_nfor which a_n e K is absolutely summable (5.3). Let ^CK — Z lnl- Then, for each function /e «^T(X; K), the series ]T t_nf(a_n)* is absolutely convergent, because the only nonzero terms are those for which a_n e K, and we have £ \tJ(a_n)\ S II/H • E t_n = c_K-\\f\\. a_n e K a_n £ K This also shows that/i->£ ^/fe) is a measure on X. This measure is said to n be defined by the masses t_n at the points a_n for all n (cf. (13.18.8)). (13.1.4) Let/e JT_CW- F°^r ^each interval [a, b] containing the support of/, the value of the integral /(/) dt (8.7) is the same, and we denote it by /»+ oo /+ oo f(f) dt.* The mapping/r-^ /(/) dt is a linear form on JT_COR)> ^an(i it is J ~~ oo J — oo a measure because, for each compact interval K = [a, b~] in R and each func- tion/e jf(R; K) we have + 00 _J(t)i by the mean value theorem (8.7.7). This measure is called Lebesgue measure on the real line R. (13.1.5) Let fj. be a measure on X and let g 6 #_c(^x)- Then for each function /e X(X) it is clear that #/e jf (X), and the mapping/H-»JU(#/) is therefore