Упражнения 3) и 4) проливают свет на тот факт, о котором говорилось в конце п.4.
Пусть
![](data:image/png;base64,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)
непрерывна, а
![](data:image/png;base64,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)
имеет ограниченное изменение в промежутке
![](data:image/png;base64,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)
.
Опираясь на оценку (25), доказать непрерывность интеграла Стилтьеса
![](data:image/png;base64,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)
по переменному верхнему пределу
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACKSURBVDjLY2hsbGQgFTMMIU0N0cY+DAwM/xkYZN945NUboojJeOxJ6+jgwWlTjptsMYNx1EIYTZTz6vM8DGUZGN7IuuUUE+2njo40Hg8Zhj3G0Q0+RGvKczNOMTGRXYHNaVg1gTwOchbYibIeK0GeBxkCCxgUTWBPg0IJajrMiSAxdL8NpRRBLAYA56kdmBJMfxIAAAAASUVORK5CYII=)
в точке
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADGSURBVDjLY2hsbGSgFDOMGjJqCKmGNEQb+zAwMPxnYJB945FXb4giJuOxJ62jg4dol+S4yRYzGEcthNFkeac+z8NQloHhjaxbTjHZYdLRkcbjIcOwxzi6wYdsQ/LcjFNMTGRXoHsF7D1g+KC7EMMAUECCFIG9JOuxEhSYIEPd4t0SZY2jekFqooxle2EBj2IIzBaY7TAvwWwGycO8B7OI5HQCcg3MdpArjd3yUkg3xEXWyjy2Xg1sSKy5mqxLntVoBhwKhgAA0Hlay96NnisAAAAASUVORK5CYII=)
, где функция
![](data:image/png;base64,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)
непрерывна.
Заключение сразу вытекает из неравенства
![](data:image/png;base64,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)
если принять во внимание, что в точке
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADGSURBVDjLY2hsbGSgFDOMGjJqCKmGNEQb+zAwMPxnYJB945FXb4giJuOxJ62jg4dol+S4yRYzGEcthNFkeac+z8NQloHhjaxbTjHZYdLRkcbjIcOwxzi6wYdsQ/LcjFNMTGRXoHsF7D1g+KC7EMMAUECCFIG9JOuxEhSYIEPd4t0SZY2jekFqooxle2EBj2IIzBaY7TAvwWwGycO8B7OI5HQCcg3MdpArjd3yUkg3xEXWyjy2Xg1sSKy5mqxLntVoBhwKhgAA0Hlay96NnisAAAAASUVORK5CYII=)
должна быть непрерывна и вариация
![](data:image/png;base64,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)
.
Если
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAATCAYAAABPwleqAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABsSURBVDjLY2hsbGQgFzMMM831eR6GsgwMbxgYGP7DsHF0gw9BzWCNsh4r0zo6eFAMMo5aSFhzfaykEYPR7tj6eknynA00wJiB4a6sW04xyZqjjBkWMsganTKSZTjFIOOxB+aF4RzPo5oHm2YAfMGoZyMW3koAAAAASUVORK5CYII=)
есть класс непрерывных в промежутке
![](data:image/png;base64,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)
функций, а
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACKSURBVDjLY2hsbGQgFTMMnKaGaGMfBgaG/yhYxmNPWkcHD1ZNYA3GUQs7OtJ4PGRlV3rk1RsStCkrK0sFRNfneRjKynqsRDadoJ+ijBkWGkc3+BDtp/r6WEkjBqPdsfX1kqQFBNBfJIUeyGmybjnFIBuNjaPridKU4yZbDAlq47u4nMgwxJIRMRgAPscfBNViJQYAAAAASUVORK5CYII=)
- класс функций с ограниченным изменением в этом промежутке, то, как известно, каждая функция одного класса, интегрируема по каждой функции другого класса. Доказать, что ни один, ни другой из этих классов не может быть расширен с сохранением упомянутого свойства.
Это, ввиду 4), почти очевидно относительно класса
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAATCAYAAABPwleqAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABuSURBVDjLY2hsbGQgFzMMM831eR6GsgwMbxgYGP7DsHF0gw9BzWCNsh4r0zo6eFAMMo5aSFhzfaykEYPR7tj6eknynA00wJiB4a6sW04xyZqjjRkqGWSNZhjJMsxgkPXoLe/o4Bzu8TyqebBpBgDS+qe8YB6SFQAAAABJRU5ErkJggg==)
. Действительно, если функция
![](data:image/png;base64,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)
имеет точку разрыва
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAYCAYAAAAcYhYyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADGSURBVDjLY2hsbGSgFDOMGjJqCKmGNEQb+zAwMPxnYJB945FXb4giJuOxJ62jg4dol+S4yRYzGEcthNFkeac+z8NQloHhjaxbTjHZYdLRkcbjIcOwxzi6wYdsQ/LcjFNMTGRXoHsF7D1g+KC7EMMAUECCFIG9JOuxEhSYIEPd4t0SZY2jekFqooxle2EBj2IIzBaY7TAvwWwGycO8B7OI5HQCcg3MdpArjd3yUkg3xEXWyjy2Xg1sSKy5mqxLntVoBhwKhgAA0Hlay96NnisAAAAASUVORK5CYII=)
, то она заведомо не интегрируема, например, по функции с ограниченным изменением
![](data:image/png;base64,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)
, имеющей ту же точку разрыва.
Пусть теперь
![](data:image/png;base64,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)
в промежутке
![](data:image/png;base64,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)
имеет бесконечное полное изменение; в этом предположении построим такую непрерывную функцию
![](data:image/png;base64,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)
, для которой интеграл (30) не существует.
Если разделить промежуток
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACMAAAAXCAYAAACBMvbiAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHLSURBVEjH7ZVPTsJAFMY/YIvGLcrgliXMXIGUJuiyCaXp0i6IYQEmXZjYsmPlFfiz5AgCF/AEBvAAGo8g4gy1WEgt0EDQyOIlbTrz3m/m+94r6vU63OgCUQAxJ7pR77dtxhSI+NVZWFRiaPN1E77oHazU3BVMWSZVUYMTffA67R9hiHRd2xXEcswO/+9gvv03jewVptEw4nmCB+EJIleqv0KmEiX3csXK7h3GNPVTCtrTLSuxVZjFuSCeV88gS2WXYFpLCZhbG8MII5al85vZ7AF5lSR2ByL3jEYjHiwROkzTrhgw4nsnSEoDwzSPQsOYhnKcT0b67gC0LD3Bkz+vktKRKDIG6FDIZFXkLAHemGZfhIIRchRptOOdxKJL5CT6TLcL60g0f/86RGiYWUKwsdeAt+UcI6CPQaZ083iB+T6aQupJrtmZUDDLCw0umZTEQJxY3Nq6XTT1ueGNYTSGFmixIzrBNJUzYUTeRlNhSlV1ZgfmXabE3By2zgpRprbd7nPM73gnNIxI6nSQCDpUbTVDgaE3sV2TMynghcs58hbjeZrzvQF//0AYlSGdyFXIpgPONpQTxQhudb8Q9cDUtC/MvuMAc4D50zCfn25rkAEImfIAAAAASUVORK5CYII=)
пополам, то хоть в одной из половин полное изменение функции
![](data:image/png;base64,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)
тоже будет бесконечно; разделим эту половину снова пополам интеграл т.д. По этому методу определится некоторая точка
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAPCAYAAADQ4S5JAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAB5SURBVCjPY2hsbGQgBTMMAQ0dHWk8HjIMexgYGP6DsHF0gw9ODfV5HoayDAxvZN1yikH8HDfZYhgbQ0N9faykMQPDXXQFOJ3UEG3sw8BgfDe2vl6SKA0g6xmMoxYS7WmwBiQbwHwZjz1pHR08WDWghw4uvwyFmCaEAYTCCCQoz74zAAAAAElFTkSuQmCC)
, в каждой окрестности которой
![](data:image/png;base64,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)
не имеет ограниченного изменения. Для простоты пусть
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAATCAYAAAD4f6+NAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC/SURBVEjHY2hsbGQYTHhQOWbUQYPOQR0daTweMgx7GBgY/su65RQPmhCKMpbt9cirNxwUDqqvj5U0YjDaHVtfLzkoHNQQbezDYBy1kOg0hBzPIGwc3eBD3ehiWGgcG5tizMBwF2yHjMeetI4OHqwOqs/zMJRlYHgDS3A5brLF2BIfyFCYg7FiHCEAii6IQ4zvgqIMZh+yp5EVCwMVL5N3ywujWXRFGlsyGEdXottpHNlgOVoOjTpo1EGjDhosGABI8fn1OjF+OAAAAABJRU5ErkJggg==)
.
В таком случае легко построить последовательность возрастающих интеграл стремящихся к
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAATCAYAAABLN4eXAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABESURBVDjLY2hsbGQgFTMMIU0dHWk8HjIMexgYGP7LuuUUk2RTlLFsr0devSHRmurrYyWNGIx2x9bXSw6H0BvVNLw0AQBK5nNh9YvxIQAAAABJRU5ErkJggg==)
значений
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACsAAAAYCAYAAABjswTDAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADGSURBVFjDY2hsbGQYKnjIOHTUsaOOHXXsqGOHg2Mboo19GBgY/jMwyL7xyKs3RBGT8diT1tHBQy2Lc9xki2F2ubkZ1xJjPm6DjKMWwmhcmqOMGRZCLMSBsejt6Ejj8ZBh2AOTq6+PlTRmYLgr65ZTTFYyqM/zMJRlYHhDjAGkYrAHkTwBc7xxdIMPWY7t6Cjn9JBl6DWObLCkpkMbIo0tGRiMl8XW1wsjAsZHVZbBaAay2GhpMOrYUceOOnbUsaOOHXXsgGEAje0IoikFbu0AAAAASUVORK5CYII=)
:
![](data:image/png;base64,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)
так, чтобы ряд
![](data:image/png;base64,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)
расходился. Для этого ряда затем можно подобрать такую последовательность стремящихся к 0 чисел
![](data:image/png;base64,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)
, чтобы и ряд
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIMAAAAtCAYAAACNi9j4AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAO9SURBVHja7Zu9cdswFMcxgBfwneEuA0jYgabvvICkUxWdK58a1z5anYp4AXcuM0M8gTdIoQ20gyOQBglCAAiAIGhF/+Ld2bkEeO/hh/cFhmw2GwKBcIETIIABAhgggAECGCDxYHhesDtCyCe9vf3FCNnxn9ni+Q6OO0MYFnn+8367vXjI6CPNHh632/uLjGVP/M/gvDOFYc7IG48IgAFp4pOw+RsHAmkCBSQEMEAAw4lKUSwvGVsUQ62/YKxYFsXlqdrnov/3o/NQl4Q4akqmf4Y8rBR7DLm3bg3V1/UPvH0si8UA4W3nmDCkurVDR5+h7VP1N8LQ6h64XOXvXS1kuTghuzFhKNb5hLL5S6qDmTP6kq+LSbKoENk+WX8rDEdAELbrIpIry7L1ypvQr4mmCpIvDGIGkupwSv8cWu108MW1T9a/E4aaRkL2FRB0b7sJvqGzyl30g69ZKqZEIB8Y5LWS3lQy/UiVlmLbJ+vvBIN6e7uA8CXdlla8YEh4MGMAOIR9sv7OMHDh4+f8iryLtNE3XFUpyJ56fGBwWW+QIvJwSVKkpiHsk/X3gkEHREixqK5hM1BVsF3DtDuYLmc1HRLdZxl7cimKU8OQ2r5eMMjhvW/h5LKGrGDljCZF6X8/dlYN39deMbuemDCMYV9vGMoFrm9e+9wqVycKBYXBsoGqc0zOUqETa8U4QJ0d4hHPKJoLMJZ9/dMEpb/7Fky6Ykj3JK7CIIxWf2/WpH9l3XQOVPe2PcU3KU0fnmMVkGPZF1xAxuxzy/zmkNNkBdst7vEN0x2M6db4pDj+b26u2asWhogV/hj2BbWW4gBj5FldSKwNs8wZXNKTCqvsLLk9ZsvlajarnKruKz7icYEh5tAplX29hk7RjdaEO7GHGnVkBeX3EhOU6rhWnaDOitmkclg7F8v7+sAQcxydyr5N6Dg6pGC0bV4brQFL9wAjFFSHU7XjNOv4HhDfd1asf7Tb3ablNcEQ86FqaPtM9Y7zQ1VowahTVgBgyrEmx3IFTVW0KVr5HJLu77pGhlivh6ntM+nfCYNvwVgRbTDMMmCqP8M3PFRp2zVL2nJNa7p9VRjqvaW8G/vBKKV9Jv2tMIR/0xBvZBryPQMkjq87x6JOEmHMCxi+EQwQCJwAAQwQRxhc/hudKDZjfv8IOcHIIE/FUn8gCkkMg222LaKC6Fl5F4Lo8B/D0DVeXmdsJaJByNfRkBOCwTi7/4oWiAxnAoOYbR9/t9jAgJrhTGAwzba1dQW6CXQTEMAAAQyQc5F/+U4DVYCAmhQAAAAASUVORK5CYII=)
(31)
все же расходился. Теперь определим функцию
![](data:image/png;base64,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)
, полагая
![](data:image/png;base64,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)
а в промежутках
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADUAAAAYCAYAAABa1LWYAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADoSURBVFjDY2hsbGQYbhiFE2XMsJCBgeE/GBtHLRzMDs9xky3G5VYMT8m65RQPpVgBR8Sop0Y9NeqpUU+NeopYTyHqBNk3bm7GtQwyHnvSOjp4KKtfKDeLLE91dKTxeMgw7IFpqq+PlTRmYLhLTmxS0yyKPIWuAeYw4+gGH0otp8Qssj3VEG3sw8BgfDe2vl4SJl+f52Eoy2B0ClmMGExNsyjyFK6QRRdzM3arJZQnKDELrhYtUCj2FCz9gzK4cWxsSmRkvSG5oUmOWSCPucsZzyLLU8Oy6zHqqVFPjXpq1FOjnhr11CDDAPRhW2FIg2viAAAAAElFTkSuQmCC)
считая
![](data:image/png;base64,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)
линейной:
![](data:image/png;base64,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)
Очевидно,
![](data:image/png;base64,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)
будет непрерывна. В то же время, ввиду расходимости ряда (31), при
![](data:image/png;base64,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)
и
![](data:image/png;base64,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)
так что интеграл от
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAVCAYAAABPPm7SAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGSgBDMMDQPq62MljRkY7jIwMPyXdcspJskAkGYjBtlTHnn1hg3Rxj4MMh570jo6eIg2IMqYYSG6rUS7AGwjg/Hd2Pp6SZIM6OhI4/GQYdgD8jME4zaEoPMZjKMWkhULsJA3jm7wIc+APA9DWQajU/j8j9eAHDfZYvQoIzkQ8UUfXgMgzpe9CUo8ZBkAdj6B0McwAKaJ2MDDMACS6vAnmiFcHtDUAACYne+XddoPvAAAAABJRU5ErkJggg==)
по
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABRSURBVDjLY2hsbGQgFzMMLs0N0cY+DAwM/9GxrFtOMV7NEI2ybzzy6g2x8XFqXrUqlNlFkkFf0iVPFm5YrLEwA4OxucfEiWyDPMBGNY9qRsYAdsJ6GhRBtGIAAAAASUVORK5CYII=)
действительно не существует.
Доказанное утверждение можно сформулировать и так: если интеграл Стилтьеса (30) для данной функции
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAVCAYAAABPPm7SAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGSgBDMMDQPq62MljRkY7jIwMPyXdcspJskAkGYjBtlTHnn1hg3Rxj4MMh570jo6eIg2IMqYYSG6rUS7AGwjg/Hd2Pp6SZIM6OhI4/GQYdgD8jME4zaEoPMZjKMWkhULsJA3jm7wIc+APA9DWQajU/j8j9eAHDfZYvQoIzkQ8UUfXgMgzpe9CUo8ZBkAdj6B0McwAKaJ2MDDMACS6vAnmiFcHtDUAACYne+XddoPvAAAAABJRU5ErkJggg==)
существует по любой
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABRSURBVDjLY2hsbGQgFzMMLs0N0cY+DAwM/9GxrFtOMV7NEI2ybzzy6g2x8XFqXrUqlNlFkkFf0iVPFm5YrLEwA4OxucfEiWyDPMBGNY9qRsYAdsJ6GhRBtGIAAAAASUVORK5CYII=)
из
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACKSURBVDjLY2hsbGQgFTMMnKaGaGMfBgaG/yhYxmNPWkcHD1ZNYA3GUQs7OtJ4PGRlV3rk1RsStCkrK0sFRNfneRjKynqsRDadoJ+ijBkWGkc3+BDtp/r6WEkjBqPdsfX1kqQFBNBfJIUeyGmybjnFIBuNjaPridKU4yZbDAlq47u4nMgwxJIRMRgAPscfBNViJQYAAAAASUVORK5CYII=)
, то
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAVCAYAAABPPm7SAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGSgBDMMDQPq62MljRkY7jIwMPyXdcspJskAkGYjBtlTHnn1hg3Rxj4MMh570jo6eIg2IMqYYSG6rUS7AGwjg/Hd2Pp6SZIM6OhI4/GQYdgD8jME4zaEoPMZjKMWkhULsJA3jm7wIc+APA9DWQajU/j8j9eAHDfZYvQoIzkQ8UUfXgMgzpe9CUo8ZBkAdj6B0McwAKaJ2MDDMACS6vAnmiFcHtDUAACYne+XddoPvAAAAABJRU5ErkJggg==)
необходимо принадлежит
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAATCAYAAABPwleqAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABuSURBVDjLY2hsbGQgFzMMM831eR6GsgwMbxgYGP7DsHF0gw9BzWCNsh4r0zo6eFAMMo5aSFhzfaykEYPR7tj6eknynA00wJiB4a6sW04xyZqjjRkqGWSNZhjJMsxgkPXoLe/o4Bzu8TyqebBpBgDS+qe8YB6SFQAAAABJRU5ErkJggg==)
; аналогично, если этот интеграл по данной функции
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAA2SURBVDjLY2hsbGQgFzMMLs0N0cY+DAwM/9GxrFtOMV7NEI2ybzzy6g2x8Qexn0c1j2qmmmYAlul9D+Aht0EAAAAASUVORK5CYII=)
существует для любой
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAVCAYAAABPPm7SAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACwSURBVDjLY2hsbGSgBDMMDQPq62MljRkY7jIwMPyXdcspJskAkGYjBtlTHnn1hg3Rxj4MMh570jo6eIg2IMqYYSG6rUS7AGwjg/Hd2Pp6SZIM6OhI4/GQYdgD8jME4zaEoPMZjKMWkhULsJA3jm7wIc+APA9DWQajU/j8j9eAHDfZYvQoIzkQ8UUfXgMgzpe9CUo8ZBkAdj6B0McwAKaJ2MDDMACS6vAnmiFcHtDUAACYne+XddoPvAAAAABJRU5ErkJggg==)
из
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAATCAYAAABPwleqAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABuSURBVDjLY2hsbGQgFzMMM831eR6GsgwMbxgYGP7DsHF0gw9BzWCNsh4r0zo6eFAMMo5aSFhzfaykEYPR7tj6eknynA00wJiB4a6sW04xyZqjjRkCGeSNU43lGVIZ5D3iJk6cyDbc43lU82DTDAB1jKcwN0vx/wAAAABJRU5ErkJggg==)
, то
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAARCAYAAAACCvahAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAABRSURBVDjLY2hsbGQgFzMMLs0N0cY+DAwM/9GxrFtOMV7NEI2ybzzy6g2x8XFqXrUqlNlFkkFf0iVPFm5YrLEwA4OxucfEiWyDPMBGNY9qRsYAdsJ6GhRBtGIAAAAASUVORK5CYII=)
необходимо принадлежит
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACKSURBVDjLY2hsbGQgFTMMnKaGaGMfBgaG/yhYxmNPWkcHD1ZNYA3GUQs7OtJ4PGRlV3rk1RsStCkrK0sFRNfneRjKynqsRDadoJ+ijBkWGkc3+BDtp/r6WEkjBqPdsfX1kqQFBNBfJIUeyGmybjnFIBuNjaPridKU4yZbDAlq47u4nMgwxJIRMRgAPscfBNViJQYAAAAASUVORK5CYII=)
.
В первой теореме о предельном переходе под знаком интеграла Стилтьеса мы поставили требование, чтобы последовательность функций
![](data:image/png;base64,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)
стремилась к предельной функции
![](data:image/png;base64,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)
равномерно. Можно, однако, заменить это требование более общим условием, что эти функции ограничены в их совокупности:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANkAAAAXCAYAAACGXOqyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAO0SURBVHja7Vm9ThtBEN4WiY4GJC9UriN7RUFDuTklrpHOlpXGQinQpqBAFGTt7oq4R3Q0kXiGIKVKwxuk4A14B+K9Y3zr8e55z9ydSTLFV/j2Z3Z+vpnZNZtMJoxAINQHMgKBsAmSaRV1olgfk4EIhHDoODqOlO6sJJkhGOfR3WmSbJPhCIRwJMnpdsT5HSba0iQp5BURjEBYn2iYQwsTxgPR4/LsHC/sC3bLGHtmTDwOtd5zbW7WZnPYs2sPwl+QhVvs/q35b36uVnTvS/5aD/cEY48Qf2Iw7hW2dSXnl8WZ5Of2nohMfOrqKXMS8SfXeNpiMvZUREJCmHOY6N9uUkZRDDR+x7HJ4CGZIeH7fXEDcZfqV0Acs2eXdX8szq82blM+iP50iWRYuLuaLZMsVVLs3xiS/YsVzBhMSDWqPfBNINVIsBAZq2Jgo8nHQzL82DCvfB49x0rt2vtkOvOHKhMLtiOqRt0Hl4HNIinlF5NVcIZQUoykFFd1lN23QDDO+O+6Mju04XUmpzIy0m6l5kpaNcm8OgfqgatOdSTLibtoYI8iZkwMhyNMMsjydZTcojsDJjQeg4DK29hsvuvegftz+J7fQ8P79nm1mFX8NPE47GmfoUxSyu+8eTcx/4bkrCvD6Ax+LmrRlq8Jfvs2TTI4Q4jedb2k4zMEkWwghI61apuFYDx4RVFjtVtUnt0PKB549gBngmxjdEykxd959YHSHeu4A9nFXm/fUdJkYjnHnDckWHCLAsR1rbUfiNap/HDeorvVOjLyZJMlS7BryCNCkX2r8H8ZkoUSJ0+IxQ96jZAsNb4YaFyGTZtoDBrqjNdefl1OA2XsMZgP54EWyCQKY0j8X4avTS7Tq+P2JDSbrtMu4qRSVbuIW0Vsx1XrfPZtupKVfbgBslVZeUuTzM7uEEz2Y0DdrWKWld37u8ZwJTNnbrVav0AHV6Zz6R7qWP8Z3Pfb1zq6TDtURkbaKqKrQMh9NMS+TZEMP52XIUQjJNOq154FxvUsMHZQq/gJvil5cMJ491oI+fkiSbaycXbJxOAyRHg6l7GfXjj2SWV69s/OE03hLHMZL99m2Xhnlo2/H0h1Yq+B34Mo+mDrAd+T5GIr4mxqryvUyTofrA21SaiueSCJj4eH/Os6+/tkZHYS32zfh/h1lX2r8L/Pz0vJLhZHtlwTz72hfhcalyIeH1VHMhMDXPeUbi+QjEAg1AMyAoFAJCMQiGQEAoFIRiAQyQgEIhmBQCCSEQhEMgLhf8QfJ07d4BDJ7BUAAAAASUVORK5CYII=)
(Только при этом нужно ещё наперед предположить непрерывность предельной функции
![](data:image/png;base64,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)
).
При доказательстве достаточно рассмотреть случай, когда
![](data:image/png;base64,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)
возрастает в строгом смысле. Но для этого случая можно воспользоваться преобразованием, проведенным в п.:
![](data:image/png;base64,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)
и, имея дело уже с римановыми интегралами, просто применить теорему Арцелла.
Укажем, в заключение, другую трактовку понятия интеграла Стилтьеса, связав его с понятием аддитивной функции от промежутка.
Пусть для каждой части
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAXCAYAAAB50g0VAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADoSURBVEjHY2hsbGQYzBiFE2XMsJCBgeE/GBtHLaSnQ3LcZIux2Y3hQFm3nOKBDDFwII06cKg7ECWtAzGynQPqwPr6WEljBoa7yHZAMofsG4+8esMBdyC65SDcEG3sw8BgfDe2vl6SbAfW53kYyjIwvAFHiYzHnrSODh5soUFM6BlHN/jAxDo60ng8ZBj2IIvhdWC0MUOgrEueFYrBseZqQB/mp5WX88LUmMfWq4FoBuPoQKKjF2SOvEfcxIkT2cB6GRgaGBjky1zKISGH7AZkcwd1LTLqwFEHjjpw1IGjDhx14KgDBx4DAPrwUPhVK+1GAAAAAElFTkSuQmCC)
данного промежутка
![](data:image/png;base64,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)
определено число
![](data:image/png;base64,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)
, причем, если промежуток
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAXCAYAAAB50g0VAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADoSURBVEjHY2hsbGQYzBiFE2XMsJCBgeE/GBtHLaSnQ3LcZIux2Y3hQFm3nOKBDDFwII06cKg7ECWtAzGynQPqwPr6WEljBoa7yHZAMofsG4+8esMBdyC65SDcEG3sw8BgfDe2vl6SbAfW53kYyjIwvAFHiYzHnrSODh5soUFM6BlHN/jAxDo60ng8ZBj2IIvhdWC0MUOgrEueFYrBseZqQB/mp5WX88LUmMfWq4FoBuPoQKKjF2SOvEfcxIkT2cB6GRgaGBjky1zKISGH7AZkcwd1LTLqwFEHjjpw1IGjDhx14KgDBx4DAPrwUPhVK+1GAAAAAElFTkSuQmCC)
точкой
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA0AAAARCAYAAAAG/yacAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAA5SURBVDjLY2hsbGQgFTNQRVN9no+qLAPDRgYG42Wx9fXCYLH6WGFjBoZl8m55YdSzaVTTqKbBrAkAqRVLdUVhku4AAAAASUVORK5CYII=)
разложен на части
![](data:image/png;base64,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)
и
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACcAAAAXCAYAAACI2VaYAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACjSURBVEjHY2hsbGQYrBiFE2XMsJCBgeE/GBtHLaSXI3LcZIux2YvhOFm3nOKBCilw4Iw6bqg5DiVNAzG6XQPiuPr6WEljBoa7yGZDMoHsG4+8esMBdRy6pSDcEG3sw8BgfDe2vl6SbMfV53kYyjIwvEE2CFtIEAo14+gGH5hYR0caj4cMwx5kMYKOG9SF8KjjRh036rhRx406btRxo44b3o4DAEICOBn8i/nfAAAAAElFTkSuQmCC)
, то и