Vektorna mira aditivna funkciya mnozhin viznachena na kilci mnozhin zi znachennyami v normovanomu prostori Ye uzagalnennyam ponyat miri zaryadu i kompleksnoyi miri Dlya vektornih mir yak i dlya mir viznacheno ponyattya integralu Zmist 1 Oznachennya 1 1 Variaciya i napivvariaciya 1 2 Vlastivosti 2 Prikladi 3 Div takozh 4 LiteraturaOznachennya red Yaksho F displaystyle mathcal F nbsp ye algebroyu mnozhin a E displaystyle E nbsp normovanim prostorom to funkciya n F E displaystyle nu colon mathcal F to E nbsp sho zadovolnyaye umovu n A B n A n B displaystyle nu A cup B nu A nu B nbsp dlya vsih mnozhin A B F displaystyle A B in mathcal F nbsp sho mayut porozhnij peretin nazivayetsya vektornoyu miroyuYaksho M displaystyle mathfrak M nbsp ye s algebroyu to funkciya n M E displaystyle nu colon mathfrak M to E nbsp nazivayetsya zlichenno aditivnoyu s aditivnoyu vektornoyu miroyu yaksho dlya kozhnoyi poslidovnosti A n n N displaystyle A n n in mathbb N nbsp mnozhin iz M displaystyle mathfrak M nbsp sho poparno ne peretinayetsya n n 1 A n n 1 n A n displaystyle nu left bigcup n 1 infty A n right sum n 1 infty nu A n nbsp Variaciya i napivvariaciya red Nehaj n F E displaystyle nu colon mathcal F to E nbsp ye vektornoyu miroyu a P F displaystyle Pi subset mathcal F nbsp poznachaye rizni skinchenni pidmnozhini iz F displaystyle mathcal F nbsp i dlya kozhnoyi P displaystyle Pi nbsp yiyi elementi poparno ne peretinayutsya i P P A textstyle bigcup P in Pi A nbsp Funkciya n F 0 displaystyle nu colon mathcal F to 0 infty nbsp zadana yak n A sup P P n P P displaystyle nu A sup left sum P in Pi nu P colon Pi right nbsp nazivayetsya variaciyeyu vektornoyi miri n displaystyle nu nbsp Funkciya n F 0 displaystyle nu colon mathcal F to 0 infty nbsp zadana yak n A sup x n A x E x 1 displaystyle nu A sup x star circ nu A colon x star in E star x star leqslant 1 nbsp nazivayetsya napivvariaciyeyu vektornoyi miri n displaystyle nu nbsp Vektorna mira n displaystyle nu nbsp maye skinchennu variaciyu yaksho yiyi na usomu prostori ye skinchennoyu Vlastivosti red Yaksho M displaystyle mathfrak M nbsp ye s algebroyu pimnozhin M displaystyle M nbsp a n M R displaystyle nu colon mathfrak M to mathbb R nbsp ye zlichenno aditivnoyu funkciyeyu mnozhin do n n n n displaystyle nu nu nu nu nbsp de n n displaystyle nu nu nbsp ye vidpovidno dodatnoyu i vid yemnoyu variaciyami Variaciya vektornoyi miri ye aditivnoyu funkciyeyu mnozhin Variaciya zlichenno aditivnoyi vektornoyi miri ye miroyu Napivvariaciya vektornoyi miri ye subaditivnoyu ta monotonnoyu funkciyeyu mnozhin Yaksho n displaystyle nu nbsp ye vektornoyu miroyu to n n displaystyle nu leqslant nu nbsp Vektorna mira z obmezhenoyu variaciyeyu ye zlichenno aditivnoyu todi j lishe todi koli yiyi variaciya ye zlichenno aditivnoyu Nehaj M s F displaystyle mathfrak M sigma mathcal F nbsp s algebra porodzhena algebroyu F displaystyle mathcal F nbsp Yaksho n M E displaystyle nu colon mathfrak M to E nbsp ye zlichenno aditivnoyu vektornoyu miroyu z obmezhenoyu variaciyeyu to dlya kozhnogo A F displaystyle A in mathcal F nbsp vikonuyetsya rivnist n F A n A displaystyle nu mathcal F A nu A nbsp Yaksho variaciya vektornoyi miri n displaystyle nu nbsp ye skinchennoyu miroyu to n displaystyle nu nbsp ye zlichenno aditivnoyu vektornoyu miroyu Mnozhina znachen s aditivnoyi vektornoyi miri ye obmezhenoyu Prikladi red Zlichenno aditivna vektorna mira Nehaj T L 0 1 X displaystyle T colon L infty 0 1 to X nbsp ye neperervnim linijnim operatorom Todi mozhna vvesti skinchenno aditivnu miru znachennya yakoyi llya kozhnoyi vimirnoyi u sensi Lebega mnozhini A 0 1 displaystyle A subset 0 1 nbsp ye rivnim n A T x A displaystyle nu A T chi A nbsp dd de x A displaystyle chi A nbsp harakteristichna funkciya Takozh dlya kozhnogo A 0 1 displaystyle A subset 0 1 nbsp n A l A T displaystyle nu A leqslant l A T nbsp de l displaystyle l nbsp mira Lebega dd Todi takozh n A l A T displaystyle nu A leqslant l A T nbsp sho dovodit sho n displaystyle nu nbsp ye vektornoyu miroyu iz skinchennoyu variaciyeyu Vektorna mira iz skinchennoyu napivvariaciyeyu ale neskinchennoyu variaciyeyu Nehaj L 0 1 displaystyle mathcal L 0 1 nbsp ye s algebroyu pidmnozhin Lebera mnozhini 0 1 displaystyle 0 1 nbsp Funkciya n L 0 1 L 0 1 displaystyle nu colon mathcal L 0 1 to L infty 0 1 nbsp zadana yakn A x A displaystyle nu A chi A nbsp dd dlya A L 0 1 displaystyle A in mathcal L 0 1 nbsp maye skinchennu napivvariaciyu ale neskinchennu variaciyu Vektorna mira iz neskinchennoyu variaciyeyu Nehaj F A N A lt ℵ 0 N A lt ℵ 0 displaystyle mathcal F A subseteq mathbb N colon A lt aleph 0 vee mathbb N setminus A lt aleph 0 nbsp Funkciya n F R displaystyle nu colon mathcal F to mathbb R nbsp zadana yakn A A A lt ℵ 0 A N A lt ℵ 0 displaystyle nu A left begin array ll A amp A lt aleph 0 A amp mathbb N setminus A lt aleph 0 end array right nbsp dd maye neobmezhenu variaciyu Div takozh red Zaryad teoriya miri Kompleksna mira Mira mnozhiniLiteratura red Cohn Donald L 1997 Measure theory vid reprint Boston Basel Stuttgart Birkhauser Verlag s IX 373 ISBN 3 7643 3003 1 Zbl 0436 28001 Arhiv originalu za 28 sichnya 2022 Procitovano 3 lyutogo 2022 Diestel Joe Uhl Jerry J Jr 1977 Vector measures Mathematical Surveys 15 Providence R I American Mathematical Society s xiii 322 ISBN 0 8218 1515 6 Kluvanek I Knowles G Vector Measures and Control Systems North Holland Mathematics Studies 20 Amsterdam 1976 Otrimano z https uk wikipedia org w index php title Vektorna mira amp oldid 36028683
