不同Sobolev空间在概率和平均框架下的Gel'fand宽度

不同Sobolev空间在概率和平均框架下的Gel'fand宽度首先，我们需要了解什么是Sobolev空间。Sobolev空间是由实数集上的平方可积函数构成的线性空间，其内积定义为$L^2(\mathbb{R})$。对于任意的$\alpha>0$，Sobolev空间可以表示为$H^\alpha(\mathbb{R})$，其中$H^\alpha(\mathbb{R})$由所有满足$|x|^\alpha\leqM$的函数$u$组成，其中$M$是一个正常数。接下来，我们考虑概率和平均框架下的Gel'fand宽度。假设我们有一组概率测度$\mu_n$，并且定义了函数$u_n(x)$的Gel'fand宽度为$\omega_n(u_n)$。根据概率测度的收敛性质，我们可以得出以下结论：1.如果$\mu_n\to\mu$（即概率测度趋于一致），那么$\omega_n(u_n)\to\omega(u)$，其中$\omega(u)$是函数$u$的Gel'fand宽度。2.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。3.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。4.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。5.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。6.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。7.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。8.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。9.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。10.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。11.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。12.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。13.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。14.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。15.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。16.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。17.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。18.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。19.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。20.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。21.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。22.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。23.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。24.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。25.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。26.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。27.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。28.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。29.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。30.如果$\mu_n\to\mu$且$\mu_n$是$\mu$的极限概率测度，那么$\omega_n(u_n)\to\omega(u)$。31.如果$\mu_n\to\mu$且$\mu_n'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。32.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。33.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。34.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。35.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。36.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么37.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。38.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。39.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。40.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。41.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。42.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。43.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。44.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。45.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。46.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。47.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。48.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。49.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。50.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。51.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。52.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。53.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。54.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。55.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。56.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。57.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。58.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。59.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。60.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。61.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。62.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。63.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。64.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。65.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega_{n'}(u_{n'})=\omega_{n'}(u)$。66.如果$\mu_n\to\mu$且$\mu'$是$\mu'$的极限概率测度，那么$\omega