Zbizhnist v L p displaystyle L p v funkcionalnomu analizi teoriyi jmovirnostej i sumizhnih disciplinah vid zbizhnosti vimirnih funkcij abo vipadkovih velichin Zmist 1 Viznachennya 2 Terminologiya 3 Vlastivosti zbizhnosti v UNIQ postMath 00000012 QINU 4 DzherelaViznachennya red Nehaj X F m displaystyle X mathcal F mu nbsp prostir z miroyu Todi prostir L p L p X F m displaystyle L p equiv L p X mathcal F mu nbsp vimirnih funkcij takih chto yih p displaystyle p nbsp ta stepin de p 1 displaystyle p geqslant 1 nbsp integrovna za Lebegom ye metrichnim Metrika v comu prostori maye viglyad d f g f g p X f x g x p m d x 1 p displaystyle d f g f g p equiv left int limits X f x g x p mu dx right 1 p nbsp Nehaj dana poslidovnist f n n 1 L p displaystyle f n n 1 infty subset L p nbsp Todi kazhut sho cya poslidovnist zbigayetsya v L p displaystyle L p nbsp do funkciyi f L p displaystyle f in L p nbsp yaksho vona zbigayetsya v metrici viznachenij vishe tobto lim n f n f p 0 displaystyle lim limits n to infty f n f p 0 nbsp Pishut f n L p f displaystyle f n stackrel L p longrightarrow f nbsp U terminah teoriyi jmovirnostej poslidovnist vipadkovih velichin X n n 1 L p W F P displaystyle X n n 1 infty subset L p Omega mathcal F mathbb P nbsp zbigayetsya do X displaystyle X nbsp z togo zh prostoru yaksho lim n E X n X p 0 displaystyle lim limits n to infty mathbb E X n X p 0 nbsp Pishut X n L p X displaystyle X n stackrel L p longrightarrow X nbsp Terminologiya red Zbizhnist v prostori L 1 displaystyle L 1 nbsp nazivayetsya zbizhnistyu v serednomu Zbizhnist v prostori L 2 displaystyle L 2 nbsp nazivayetsya zbizhnistyu v serednokvadratichnomu Vlastivosti zbizhnosti v L p displaystyle L p red Yedinist granici Yaksho f n L p f displaystyle f n stackrel L p longrightarrow f nbsp i f n L p g displaystyle f n stackrel L p longrightarrow g nbsp to f g displaystyle f g nbsp m displaystyle mu nbsp majzhe vsyudi P displaystyle mathbb P nbsp majzhe napevno Prostir L p displaystyle L p nbsp povnij Yaksho f n f m p 0 displaystyle f n f m p to 0 nbsp pri min n m displaystyle min n m to infty nbsp to isnuye f L p displaystyle f in L p nbsp takij sho f n L p f displaystyle f n stackrel L p longrightarrow f nbsp Iz zbizhnosti v L p displaystyle L p nbsp viplivaye zbizhnist za miroyu za jmovirnistyu Yaksho f n L p f displaystyle f n stackrel L p longrightarrow f nbsp to f n m f displaystyle f n stackrel mu longrightarrow f nbsp Dzherela red Kolmogorov A N Fomin S V Elementy teorii funkcij i funkcionalnogo analiza 4 e izd Moskva Nauka 1976 544 s ISBN 5 9221 0266 4 ros Berezanskij Yu M Us G F Sheftel Z G Funkcionalnyj analiz kurs lekcij K Visha shkola 1990 600 s ros Otrimano z https uk wikipedia org w index php title Zbizhnist v Lp amp oldid 17335126
