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5 Young $BB,EY(B

$BDjM}(B 5   $\Omega(\subset{\mbox{\sl R}}^M)$ $B$r3+=89g(B, $K (\subset{\mbox{\sl R}}^N)$ $B$rM-3&=89g$H$7!"(B $v_n: \Omega\rightarrow R^N$ $B$,2DB,4X?tNs$G(B $v_n(x)\in K$ (a.e. $x\in\Omega$) $B$G$"$k$J$i$P(B $B$"$kItJ,Ns(B $\{v_{n_j}\}_j$, $B$*$h$S(B ${\mbox{\sl R}}^N$ $B>e$N$"$k3NN(B,EY(B ($BA4B,EY(B 1 $B$NHsIi(B Borel $BB,EY(B) $B$NB2(B $\{\nu_x(y)\}_{\mbox{\scriptsize a.e.}x\in\Omega}$ $B$,B8:_$7!"

$\mathop{\rm supp}_y \nu_x(y)\subset \bar{K}$

${\mbox{\sl R}}^N$ $B>e$NG$0U$NO"B34X?t(B $G(y)$ $B$KBP$7!"(B

$$G(v_{n_j}(x))\rightarrow\int_{{\mbox{\scriptsize\sl R}}^N}G(y)d\nu_x(y) \hspace{1zw}L^\infty(\Omega)\mbox{weak$\ast$}$$

$B>\:Y$O(B [13,21] $B;2>H$N$3$H!#(B

$B$3$N3NN(B,EY$NB2(B $\{\nu_x(y)\}$ $B$r(B $\{v_{n_j}\}_j$ $B$KBP$9$k(B Young $BB,EY(B $B$H$$$&!#0J8e!"(BYoung $BB,EY$K$h$k@QJ,(B

$$\int_{{\mbox{\scriptsize\sl R}}^N}G(y)d\nu_x(y)$$

($\nu$ $B$NE:;z(B $x$ $B$N4X?t$H$J$k(B) $B$r(B $\bar{G}(x)$ $B$d(B $\langle \nu_x(y),G(y)\rangle$ $B$N$h$&$K=q$/$3$H$H$9$k!#(B

$B$J$*!"(BYoung $BB,EY$,A4B,EY(B 1 $B$G$"$kI,A3@-$O(B $G(y)\equiv$$BDj?t(B $B$H$9$l$P(B $BMF0W$K$o$+$k$@$m$&!#(B

$BNc(B 6

$v_n\rightarrow v$ $B$,6/$$0UL#$G$N<}B+$G$"$k$J$i$P(B ($BNc$($P(B a.e. $B<}B+(B)

$$G(v_n)\rightarrow G(v(x))$$

$B$G$"$j!"0lJ}(B

$$G(v_n)\stackrel{\ast}{\rightharpoonup}\langle \nu_x(y),G(y)\rangle$$

$B$J$N$G(B $\langle \nu_x(y),G(y)\rangle =G(v(x))$ $B$H$J$j!"(B $B$h$C$F(B

$$\nu_x(y)=\delta_{v(x)}(y) (=\mbox{$v(x)$ $BCf?4$N(B $\delta$-$BB,EY(B})$$

$B$H$J$k!#(B

$B$J$*!"5U$K(B $\nu_x(y)=\delta_{v(x)}(y)$ $B$G$"$C$?$H$9$k$H!"(B

$$v_{n_j}\stackrel{\ast}{\rightharpoonup}\bar{v},\hspace{1zw}v_{n_j}^2\stackrel{\ast}{\rightharpoonup}\bar{v}^2$$

$B$H$J$j!"(B$\bar{v}(x)$ $B$OM-3&$J$N$G!"G$0U$NM-3&=89g(B $O(\subset\Omega)$ $B$KBP$7$F(B

$$\begin{eqnarray*} \lefteqn{\int_\Omega\raisebox{.18em}{\mbox{$\chi$}}_{O}(x)\ve... ...-2\bar{v}\raisebox{.18em}{\mbox{$\chi$}}_{O}(x)\in L^1(\Omega)) \end{eqnarray*}$$

$B$H$J$j!"$h$C$F(B ($BI,MW$J$iItJ,Ns$r $v_{n_j}\rightarrow\bar{v}$ a.e. $B$H$J$k!#(B

$BNc(B 7

$BNc(B 4 $B$Ge$2$?(B $v_n = \cos nx$ $B$KBP$9$k(B Young $BB,EY$r(B $B7hDj$9$k(B (cf.[13])$B!#(B

$\phi\in C_0(R)$ $B$H$7(B $G(v)$ $B$rO"B34X?t$H$9$k!#(B

$$\begin{eqnarray*} \lefteqn{\int_{\mbox{\scriptsize\sl R}}\phi(x)G(\cos nx)dx =... ...\infty \phi\left(\frac{z}{n}+\frac{2k\pi}{n}\right)\right\} dz \end{eqnarray*}$$

$B$HJQ7A$9$k$H(B $\phi$ $B$NCf?H$O(B $2k\pi/n < z/n+2k\pi/n < 2(k+1)\pi/n$ $B$G$"$j!"(B $\phi\in C_0(R)$ $B$J$N$G@QJ,$NCf%+%C%3$NItJ,$O(B $n\rightarrow\infty$ $B$N$H$-$K(B $\int_{\mbox{\scriptsize\sl R}}\phi dx$ $B$K<}B+$7$+$D0lMMM-3&!"$h$C$F(B Lebesgue $B$N<}B+DjM}$K$h$j(B

$$\int_{\mbox{\scriptsize\sl R}}\phi(x)G(\cos nx)dx\rightarro... ..._{\mbox{\scriptsize\sl R}}\phi(x)dx \int_0^{2\pi}G(\cos z)dz$$

$B$H$J$k!#$3$l$O(B

$$G(\cos nx)\rightarrow \frac{1}{2\pi}\int_0^{2\pi}G(\cos z)dz \hspace{1zw}L^\infty\mbox{weak$\ast$}$$

$B$r0UL#$7!"$3$N>l9g6K8B$O(B $x$ $B$K$h$i$J$$Dj?t$H$J$k!#$3$3$G(B

$$\frac{1}{2\pi}\int_0^{2\pi}G(\cos z)dz = \frac{1}{\pi}\int... ... z)dz = \frac{1}{\pi}\int_{-1}^1\frac{G(v)}{\sqrt{1-v^2}} dv$$

$B$J$N$G!"7k6I(B $\{\cos nx\}$ $B$KBP$9$k(B Young $BB,EY$O(B

$$\nu_x(v)=\chi_{(-1,1)}(v)\frac{1}{\pi}\frac{dv}{\sqrt{1-v^2}}$$

$B$9$J$o$A(B

$$\nu_x(E)=\int_{E\cap (-1,1)}\frac{1}{\pi}\frac{dv}{\sqrt{1-v^2}}$$

$B$H$J$k!#(B

$B?^(B 2: $$\frac {1}{\pi \sqrt {1-v^2}}$$ ( $\cos nx$ $B$N(B Young $BB,EY$NL)EY4X?t(B)

$B$3$N>l9g(B Young $BB,EY$O@dBPO"B3$G!"$=$NL)EY4X?t$O(B $v=0$ $B$G$NCM$,(B $B$b$C$H$b>.$5$/!"(B$v=-1,1$ $B$G$OL58B$KH/;6$9$k!#(B

$\nu_x(v)$ $B$O3NN(B,EY$G$"$k$+$i!"(B $\langle \nu_x(v),G(v)\rangle$ $B$r(B $G(v)$ $B$N(B $B=E$_IU$-$NJ?6Q$H8+$k$H!"$3$l$O(B $v=-1,1$ $B$N$H$3$m$,$b$C$H$b=E$_IU$1$,(B $B6/$/!"(B$v=0$ $B$N$H$3$m$,$b$C$H$b=E$_IU$1$,7Z$$$3$H$r0UL#$9$k!#(B $B$3$N$3$H$ONc$($P $\cos nx$ $B$N$h$&$K?6F0$7$F<}B+$7$J$$4X?t$N>l9g$O!"3F(B $x$ $B$KBP$7$F(B $B$=$N6K8B$OE@$K$J$k$N$G$O$J$/!"$b$C$H9-$,$C$?$\$s$d$j$7$?$b$N$K$J$C$F(B $B$$$F!"O"B34X?t(B $G(v)$ $B$r$+$V$;$?$b$N$OJ?6Q<}B+$H$7$F$O6K8B$r;}$D$,(B $B$=$l$O(B $G(v)$ $B$NJ?6Q$N$h$&$J$b$N$G!"(B $\cos nx$ $B$NCM$,!V$?$/$5$s!W8=$o$l$k(B $v=-1,1$ $B$NIU6a$,=E$_$,6/$/!"(B $\cos nx$ $B$NCM$,!V>/$J$/!W8=$o$l$k(B $v=0$ $B$NIU6a$N=E$_$,7Z$/$J$k!#(B $B$J$*!"(B$v=-1,1$ $B$NIU6a$K(B $\cos nx$ $B$NCM$,!V$?$/$5$s!W8=$o$l$k!"(B $B$H$$$&$N$O!"(B $BNc$($P(B $n$ $B$,<+A3?t$G$J$/;~4V$H$7$F$N $B$NCM$NJQ2=$r9M$($F$_$k$H!"(B $B%0%i%U$N79$-$N>.$5$$(B $v=-1,1$ $B$N$H$3$m$O$f$C$/$jF0$-!"(B $B79$-$NBg$-$$(B $v=0$ $B$N$H$3$m$OB.$/DL2a$9$k$3$H$,A[A|$G$-$k$@$m$&!#(B

$B7k6I!"(B(7) $B$N$?$a$K$O!"(B

$\nu_{(x,t)}(u)$ $B$r7hDj$7!"(B $\nu_{(x,t)}(u) = \delta_{\bar{u}(x,t)}(u)$ $B$G$"$k$3$H$r<($9(B

$B$H$$$&$3$H$,L\I8$G$"$k$3$H$K$J$k!#(B