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From: chiaki@kit.ac.jp (Tsukamoto Chiaki)
Newsgroups: fj.sci.math
Subject: Re: 4$BCJ3,$G%k%Y!<%0@QJ,$r9=C[$;$h!#(B
Date: Thu, 19 Feb 2009 20:39:32 +0900
Organization: Kyoto Institute of Technology
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$B9)A!Bg$NDMK\$G$9(B.

In article <e5442653-48f5-44ec-a767-ec9766034eb2@z1g2000yqn.googlegroups.com>
kyokoyoshida123 <kyokoyoshida123@gmail.com> writes:
> $B%j!<%^%s@QJ,$G$N9-5A@QJ,$NDj5A$K;w$F$^$9$M!#(B
> $B%-%A%s$H=q$1$P(Bf=$B!g$N>l9g(B,
> $B"i(B_X fd$B&L(B:=sup{$B"i(B_X gd$B&L(B;0$B!e(Bg$B!e(Bf,$B"i(B_X gd$B&L(B<$B!g(B},
> $B&L(B(supp(f))=$B!g$N>l9g(B,
> $B"i(B_X fd$B&L(B:=sup{$B"i(B_X gd$B&L(B;0$B!e&L(B(supp(g))$B!e&L(B(supp(f)),$B&L(B(supp(f))<$B!g(B}.
> $B$HDj5A$9$k$N$G$9$M!#(B

$B$-$A$s$H=q$1$P(B,

  $B"i(B_X f d$B&L(B
   := sup { $B"i(B_X g d$B&L(B | g $B$OM-3&$G(B, $B&L(B(supp(g)) < $B!g(B, 0 $B!e(B g $B!e(B f $B$r(B
                         $BK~$?$92D@QJ,4X?t(B }
                        
$B$H$J$j$^$9$,(B, $B$=$l$O(B

  $B"i(B_X f d$B&L(B
   := sup { $B"i(B_X g d$B&L(B | g $B$OM-3&$G(B, $B&L(B(supp(g)) < $B!g(B, 0 $B!e(B g $B!e(B f $B$r(B
                         $BK~$?$9C14X?t(B }

$B$H$7$F$bF1$8$3$H$G$9(B.

> g$B$O-"$G$N(Bf$B$KAjEv$9$k$o$1$G$9$M!#(B
> $B$3$N(Bg$B$N8F$SL>$O$"$k$N$G$7$g$&$+(B?
> $B$=$7$F(B,$B@QJ,CM$OM-8B$G(Bf$B$X$NC1D4A}2CB2(B{g}
> (g$B$NE:?t$O$G$O$J$$$G$7$g$&$+$iNs$G$O$J$/B2$H$G$b8@$$$^$7$g$&$+(B)
> $B$,I,$::N$l$k$N$G$9$M!#(B
> ($B$A$g$C$HK\$G8+$D$1$l$J$$$b$N$G(B)

$B$3$NDj5A$G(B $B"i(B_X f d$B&L(B < $B!g(B $B$H$J$k(B f $B$K$D$$$F$O(B,
 g_n $B!e(B f $B$H$J$k(B, $BM-3&$G(B $B&L(B(supp(g)) < $B!g(B $B$J@5CMC14X?t(B g_n $B$G(B
 L^1 $B$N0UL#$G(B f $B$K<}B+$9$k$b$N$,<h$l$^$9$+$i(B,
$B$=$NItJ,Ns$G(B f $B$K35<}B+$9$k$b$N$,<h$l$^$9(B.
$B$=$l$OC1D4A}2C$K$b<h$l$^$9(B.

$B0lHL$N(B f $B$K$D$$$F(B, f $B$K35<}B+$9$k>e$N$h$&$J(B g_n $B$r:n$l(B,
$B$H$$$o$l$k$H:$$k$+$bCN$l$^$;$s(B. X $B$,(B $B&R(B-$BM-8B$G$"$l$P(B,
 f $B$K<}B+$9$k>e$N$h$&$J(B g_n $B$O4JC1$K:n$l$^$9(B.

$B>/$74*0c$$$7$F$$$^$7$?$,(B, $B$3$l$,(B $B&R(B-$BM-8B@-$N;H$$J}$N(B
$B0l$D$G$9(B. $B8e(B, $B&R(B-$BM-8B$G$J$$$H(B, $B40A42CK!E*$JM-8B2CK!E*(B
$BB,EY$NB,EY$X$N3HD%$O0l0U$G$J$$$+$bCN$l$J$$$H$+(B,
 Radon-Nikodym $B$NDjM}$O@.N)$7$J$$$+$bCN$l$J$$(B,
$B$H$$$C$?$H$3$m$G$7$g$&$+(B.

> $B$3$l!V-$(B f arbitrary $B"i(B|f|d$B&L(B<$B!g(B, f^+(x)=max{f(x),0}$B!f(B0.
> f^-(x)min{-f(x),0}$B!f(B0 $B!W$O@5$7$/$O(B
> $B!V-$(B f arbitrary $B"i(B|f|d$B&L(B<$B!g(B, f^+(x)=max{f(x),0}$B!f(B0.
> f^-(x)=max{-f(x),0}$B!f(B0$B!W$G$7$g$&$+(B?

$BEvA3$=$&$G$9$M(B.
-- 
$BDMK\@i=)(B@$B1~MQ?t3X(B.$B4pHW2J3XItLg(B.$B5~ET9)7]A!0]Bg3X(B
Tsukamoto, C. : chiaki@kit.ac.jp