上海电机学院学报
上海電機學院學報
상해전궤학원학보
JOURNAL OF SHANGHAI DIANJI UNIVERSITY
2011年
4期
262-268
,共7页
等度可积函数序列%几乎处处收敛%依测度收敛%积分与极限交换
等度可積函數序列%幾乎處處收斂%依測度收斂%積分與極限交換
등도가적함수서렬%궤호처처수렴%의측도수렴%적분여겁한교환
equi-integrable {unction sequence%almost everywhere convergence%convergencein measure%commutation on integral and limit
分析了积分与极限交换的充要条件,引进了等度可积函数序列和几乎处处绝对连续函数序列概念。证明几乎处处收敛的可积函数列在极限函数未知是否可积的情况下,积分与极限交换的充要条件是函数列在S∞上积分等度收敛;证明可积函数列依测度收敛的情况下,积分与极限交换的充要条件是函数列在[a,b]上积分等度收敛。
分析瞭積分與極限交換的充要條件,引進瞭等度可積函數序列和幾乎處處絕對連續函數序列概唸。證明幾乎處處收斂的可積函數列在極限函數未知是否可積的情況下,積分與極限交換的充要條件是函數列在S∞上積分等度收斂;證明可積函數列依測度收斂的情況下,積分與極限交換的充要條件是函數列在[a,b]上積分等度收斂。
분석료적분여겁한교환적충요조건,인진료등도가적함수서렬화궤호처처절대련속함수서렬개념。증명궤호처처수렴적가적함수렬재겁한함수미지시부가적적정황하,적분여겁한교환적충요조건시함수렬재S∞상적분등도수렴;증명가적함수렬의측도수렴적정황하,적분여겁한교환적충요조건시함수렬재[a,b]상적분등도수렴。
The necessary and sufficient condition of the commutation on integral and limit is discussed. The concepts of equi-integrable function sequences and almost everywhere absolute continuous function sequences are introduced. The necessary and sufficient condition of commutation on integral and limit, in which the integrability of the limiting function is unknown, is proved. The necessary and sufficient condition of commutation on integral and limit, in which the integrable function sequence is convergent in measure, is proved. The conditions are that the function sequences are equi-convergent integral.
