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實變函數(shù)漫談(8)建立測度的過程1

2023-06-22 11:26 作者:南海之聲sonnet耳放 0人讀過 | 我要投稿

建立測度是這樣的一個順序：對一個基本的集類 $E$ ，比如說左開右并的區(qū)間，先定義其測度，然后可以可以將它延拓到 $R(E)$ 上，然后嘗試延拓到 $S(E)$ ，在延拓的過程中牢記保持測度的可列可加性。現(xiàn)在不管測度是如何延拓的，先從可列可加性看看測度在σ環(huán)上具有哪些性質(zhì)。測度是否對單調(diào)序列具有連續(xù)性：分別考慮單調(diào)增和單調(diào)減的情況： $m(%5Cbigcup_%7Bi%3D1%7D%5E%7B%5Cinfty%7D%20A_i)%3D%5Clim_%7Bn%5Cto%20%5Cinfty%7D%20m(A_i)%EF%BC%8Cm(%5Cbigcap_%7Bi%3D1%7D%5E%7B%5Cinfty%7D%20A_i)%3D%5Clim_%7Bn%5Cto%20%5Cinfty%7D%20m(A_i)$

如果 $A_i$ 是單調(diào)增的，那么定義 $B_i%3D%5Cbigcup_%7Bn%3D1%7D%5E%7Bi%7DA_n-%20%5Cbigcup_%7Bn%3D1%7D%5E%7Bi-1%7DA_n%E5%8F%AF%E4%BB%A5%E5%8F%91%E7%8E%B0B_i%E6%98%AF%E4%BA%92%E4%B8%8D%E7%9B%B8%E4%BA%A4%E7%9A%84$

$m(%5Cbigcup_%7Bi%3D1%7D%5E%7B%5Cinfty%7D%20A_i)%3Dm(%5Cbigcup_%7Bi%3D1%7D%5E%7B%5Cinfty%7D%20B_i)%3D%5Csum_%7Bi%3D1%7D%5E%5Cinfty%20m(B_i)%20%3D%5Clim_%7Bn%5Cto%5Cinfty%7D%5Csum_%7Bi%3D1%7D%5En%20m(B_i)%20%3D%5Clim_%7Bn%5Cto%5Cinfty%7Dm(%5Cbigcup_%7Bi%3D1%7D%5E%7Bn%7D%20B_i)%3D%5Clim_%7Bn%5Cto%5Cinfty%7Dm(%5Cbigcup_%7Bi%3D1%7D%5E%7Bn%7D%20A_i)$

如果 $A_i$ 是單調(diào)減的，定義 $B_i%3DA_1-A_i%E5%8F%AF%E4%BB%A5%E5%8F%91%E7%8E%B0B_i%E6%98%AF%E5%8D%95%E8%B0%83%E5%A2%9E%E7%9A%84$ $m(%5Cbigcap_%7Bi%3D1%7D%5E%7B%5Cinfty%7D%20A_i)%3Dm(A_1-%5Cbigcup_%7Bi%3D1%7D%5E%7B%5Cinfty%7D%20B_i)%3Dm(A_1)-%5Clim_%7Bn%5Cto%20%5Cinfty%7D%20m(B_n)%3D%5Clim_%7Bn%5Cto%20%5Cinfty%7D%20(m(A_1)-m(B_n))%3D%5Clim_%7Bn%5Cto%20%5Cinfty%7D%20A_i$