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Помогите, пожалуйста, разобраться с правилами вычисления интеграла Лебега, а так же с тем, как искать минимальную сигма-алгебру. Просто общий подход, "скелет" решения таких номеров.
Пожалуйста, не отсылайте к материалам в интернете, или к учебной литературе, я уже устал искать там нормальное объяснение. Хотелось бы услышать объяснение "для чайников", т.е. "на пальцах". Дело в том, что у нас ужасный преподаватель по функциональному анализу, и из того, что объясняется на лекциях и семинарах, усваивается в лучшем случае 1%.
В качестве примера для объяснения прошу использовать вот такие 2 номера:
Здравствуйте, Барс Иван! 1) Для абсолютно интегрируемой функции интеграл Лебега и интеграл Римана совпадают. Поэтому искомый интеграл равен [$8721$]n=0[$8734$][$8747$]n/3(n+1)/3(1/[3x]!)dx=[$8721$]n=0[$8734$][$8747$]n/3(n+1)/3(1/n!)dx=[$8721$]n=0[$8734$](1/n!)1/3=1/3[$8721$]n=0[$8734$]1/n!=e/3
2) Так как множества A и B не пересекаются, то искомая алгебра должна содержать [$8709$],A,B,A[$8746$]B. С другой стороны, набор этих четырех множеств является алгеброй с единицей E=A[$8746$]B. Так как эта алгебра является конечной, то она является [$963$]-алгеброй. Ответ: {[$8709$],A,B,A[$8746$]B}
Неизвестный
11.06.2013, 22:30
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Большое спасибо за Ваш ответ!
Только оформление не очень удачное - сразу после знака интеграла написано _n/3 ^(n+1)/3 - это пределы интегрирования? Если да, то почему они именно такие? Объясните, пожалуйста.
Это пределы интегрирования, а выбраны они так, что между ними подинтегральная функция является постоянной так, что интеграл равен значению функции, умноженному на длину промежутка интегрирования, равному 1/3.
Это пределы интегрирования, а выбраны они так, что между ними подинтегральная функция является постоянной так, что интеграл равен значению функции, умноженному на длину промежутка интегрирования, равному 1/3.
Об авторе:
Facta loquuntur.
Неизвестный
12.06.2013, 12:39
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Спасибо, с этим вроде бы разобрался, нарисовав график.
Можно еще один вопрос задать - объясните, пожалуйста, поподробнее переход от первого знака равенства ко второму. Почему 1/[3x]! заменилось на 1/n! ? Откуда у нас вообще взялась переменная n и как она связана с x?
У нас этой темы, можно сказать, не было, поэтому я понимаю только объяснения "на пальцах", Вы уж извините, если я задаю простые вопросы.
P.S. Второй номер понял, большое спасибо!
Неизвестный
12.06.2013, 12:39
общий
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Спасибо!
Неизвестный
12.06.2013, 12:54
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Всё, сам разобрался, спасибо!
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