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142 XIII INTEGRATION

(13.9.18) Let fi be a bounded positive measure on X. Then for each & > 0 there
exists a compact subset KofX such that jj.(X - K) ^ e.

This is a particular case of (13.7.9).

(1 3.9.1 9) Suppose that \JL is bounded. Let (/,) be a uniformly bounded sequence
of measurable functions (i.e., sup ||/J < + oo) such that limf_n(x) =/(*) exists

n H-+OO

for almost all x e X. Then f is integrable, and we have

(13.9.19.1) |/^=lim \f_nd/i..

J »•+» J

This is a particular case of (13.8.4).

Example

(13.9.20) Let JJL be a positive measure on X and let Y be a closed subspace
of X. It follows from the Tietze-Urysohn theorem (4.5.1 ) that every function
/e Jf _K(Y) is the restriction to Y of a function g e <%£ (X) such that ||#|| = ||/||.
The function /^Y = g<p_y , which is equal to/in Y and zero outside Y, is there-
fore ju-integrable (13.9.13), and |^(/^Y)| g ||/|| • XSupp(/)). Write ,UY(/) =
K/^Y) (which clearly depends only on/); then the discussion above shows that
/-» ju_y(/) is a positive measure on Y, which is said to be the measure induced
by IL on Y. From (12.7.8), the remarks above and (13.5.7), it follows im-
mediately that if /belongs to ,/(Y) and if as above we denote by/^Y the func-
tion which is equal to/in Y and is zero in X — Y, then we have ju*(/) = /^*(/^Y).
Hence we conclude (13.5.5) that this formula is valid for any mapping
/: Y -+ R (with/^Y defined as before). Hence, finally, a function /: Y -+ E is
/i_Y-5ntegrable if and only if /^Y is ^-integrable, and that in this case we have

(13.9.20.1)

f - f

J ^^Y r

Likewise, a mapping u of Y into a topological space Z is /*_Y-;measurable
if and only if the mapping u': X -> Z which extends u to X by giving u'(x) an
(arbitrary) constant value for x e X - Y, is /i-measurable.

Conversely, consider a closed subset Y of X and a positive measure v on
Y, and let \JL be the canonical extension of v to X (13.1.7). Clearly we have
v = u_y. Also, if/e >(Y), then the function g which is equal to/on Y and
to +00 on X - Y belongs to J^(X), and we have v*(/) = ^(g\ It follows



	142 XIII INTEGRATION (13.9.18) Let fi be a bounded positive measure on X. Then for each & > 0 there exists a compact subset KofX such that jj.(X - K) ^ e. This is a particular case of (13.7.9). (1 3.9.1 9) Suppose that \JL is bounded. Let (/,) be a uniformly bounded sequence of measurable functions (i.e., sup \|\|/J < + oo) such that limf_n(x) =/() exists* n H-+OO for almost all x e X. Then f is integrable, and we have (13.9.19.1) \|/^=lim \f_nd/i.. J »•+» J This is a particular case of (13.8.4). Example (13.9.20) Let JJL be a positive measure on X and let Y be a closed subspace of X. It follows from the Tietze-Urysohn theorem (4.5.1 ) that every function /e Jf _K(Y) is the restriction to Y of a function g e <%£ (X) such that \|\|#\|\| = \|\|/\|\|. The function /^Y = g<p_y , which is equal to/in Y and zero outside Y, is there- fore ju-integrable (13.9.13), and \|^(/^Y)\| g \|\|/\|\| • XSupp(/)). Write ,UY(/) = K/^Y) (which clearly depends only on/); then the discussion above shows that /-» ju_y(/) is a positive measure on Y, which is said to be the measure induced by IL on Y. From (12.7.8), the remarks above and (13.5.7), it follows im- mediately that if /belongs to ,/(Y) and if as above we denote by/^Y the func- tion which is equal to/in Y and is zero in X — Y, then we have ju(/) = /^(/^Y). Hence we conclude (13.5.5) that this formula is valid for any mapping /: Y -+ R (with/^Y defined as before). Hence, finally, a function /: Y -+ E is /i_Y-5ntegrable if and only if /^Y is ^-integrable, and that in this case we have

	(13.9.20.1)	f - f J ^^Y r

	Likewise, a mapping u of Y into a topological space Z is /_Y-;measurable if and only if the mapping u':* X -> Z which extends u to X by giving u'(x) an (arbitrary) constant value for x e X - Y, is /i-measurable. Conversely, consider a closed subset Y of X and a positive measure v on Y, and let \JL be the canonical extension of v to X (13.1.7). Clearly we have v = u_y. Also, if/e >(Y), then the function g which is equal to/on Y and to +00 on X - Y belongs to J^(X), and we have v(/) = ^(g\* It follows