Лекции по функциональному анализу Смолянова О.Г., страница 2

Åñëè (fn ) è (gn ) - ïîñëåäîâàòåëüíîñòè íåîòðèöàòåëüíûõ ïðîñòûõôóíêöèé, ïðè÷åì fn % f è gn % f, òîZZlimfn dν = limgn dνn→∞n→∞ΩΩ(äîêàçàòåëüñòâî áóäåò ïðèâåäåíî â ñëåäóþùåé ëåêöèè).Îïðåäåëåíèå 8. Åñëè f - íåîòðèöàòåëüíàÿ ïðîñòàÿ ôóíêöèÿ è Ω1 ⊂ Ω , Ω ∈ A. ÒîãäàZZdeff (ω) dν =Ω1γΩ1 (ω)f (ω) dν.Ω- èíòåãðàë Ëåáåãà ïî èçìåðèìîìó ïîäìíîæåñòâó.Ñâîéñòâà èíòåãðàëà Ëåáåãà îò ïðîñòûõ íåîòðèöàòåëüíûõ ôóíêöèé.(1)R¡¢RRf1 (ω) + f2 (ω) dν = f1 (ω) dν + f2 (ω)dν .Ω(2)Rcf (ω) dν = cΩRΩΩf (ω) dν, c > 0.ΩR(3) Åñëè f (u) = 0 ïî÷òè âñþäó, òî(4) Åñëè Ω =nFi=1f (ω) dν = 0 .ΩΩi è ìíîæåñòâà Ωi - èçìåðèìû, òî(5) Åñëè f1 (ω) > f2 (ω) > 0∀ω , òîRf1 (ω) dν >ΩRRf (ω) dν =n RPi=1 ΩiΩf (ω) dν .f2 (ω) dν.ΩÄîêàçàòåëüñòâî.

(5) Ðàññìîòðèì ïðåäñòàâëåíèå¡¢f1 (ω) − f2 (ω) + f2 (ω) = f1 (ω).|{z} | {z }> 0> 0Òîãäà ìîæíî çàïèñàòü òàê:ZZZZZ¡¢f2 (ω) dν 6 f1 (ω) dν.f1 (ω) − f2 (ω) dν + f2 (ω) dν = f1 (ω)dν ⇒Ω|{z}> 0 (èç îïðåäåëåíèÿ)Ω| {z }ΩΩΩ> 0¤2 ËåêöèÿÎïðåäåëèì èíòåãðàë Ëåáåãà äëÿ ïðîèçâîëüíûõ íåîòðèöàòåëüíûõ ôóíêöèé.72 ËåêöèÿÎïðåäåëåíèå 9.

Ïóñòü f (ω) > 0 ∀ω , f - èçìåðèìàÿ ôóíêöèÿ íà (Ω, A), òîãäà ∃ ïîñëåäîâàòåëüíîñòü gn (ω) ïðîñòûõ ôóíêöèé:gn (ω) % f (ω)(ñóùåñòâîâàíèå äîêàæåì ïîçæå).Ïî îïðåäåëåíèþ èíòåãðàëîì Ëåáåãà íàçûâàåòñÿ ñëåäóþùèé èíòåãðàëZZdeff (ω) dν = limgn (ω) dν.n→∞ΩΩf - íàçûâàåòñÿ èíòåãðèðóåìîé, åñëè ïðåäåë êîíå÷åí.Äîêàæåì, ÷òî ýòî îïðåäåëåíèå íå çàâèñèò îò âûáîðà ïîñëåäîâàòåëüíîñòè gn (ω).Äîêàçàòåëüñòâî.

Ïóñòü èìåþòñÿ äâå ïîñëåäîâàòåëüíîñòègn (ω) : ∀ωgn (ω) % f (ω),ϕn (ω) : ∀ωϕn (ω) % f (ω).Äîñòàòî÷íî ïðîâåðèòü, ÷òîZ∀klimZgn dν >n→∞Ωϕk (ω) dν.ΩÒîãäà îòñþäà áóäåò ñëåäîâàòü, ÷òîZZϕk (ω) dν.limgn dν > limn→∞k→∞ΩΩÒàê êàê ñèòóàöèÿ ñèììåòðè÷íà, òî áóäåò âåðíî è îáðàòíîå íåðàâåíñòâîZZlimgn dν 6 limϕk (ω) dν.n→∞k→∞ΩΩÎòñþäà áóäåò ñëåäîâàòü êîððåêòíîñòü îïðåäåëåíèÿ.Èòàê, íàì èçâåñòíî, ÷òî)nkX∀ω, gn (ω) % f (ω)⇒ ∀k, ∀ω lim gn (ω) > ϕk (ω), ϕk (ω) =λj · γAj .n→∞∀ω, ϕn (ω) % f (ω)j=1ÏóñòüAkn = {ω : gn (ω) + ε > ϕk (ω)};Cnk = {ω : gn (ω) + ε 6 ϕk (ω)};lim (gn (ω) + ε) > ϕk (ω).n→∞Ïðè ýòîì ∀ kC1k ⊃ C2k ⊃ .

. . è∞Tn=1Cnk = ∅.82 Ëåêöèÿk, òî gn+1 (ω) + ε 6 ϕk (ω), à òàê êàê gn (ω) 6 gn+1 (ω), òîÄåéñòâèòåëüíî, åñëè ω ∈ Cn+1òåì áîëåågn (ω) + ε 6 ϕk (ω), òàê ÷òî ω ∈ Cnk .Äàëåå ∀ ω ∈ Ω,∃ n0 ∈ N, òàêîå ÷òî∀ n > n0gn (ω) + ε > ϕk (ω);∞TCnk . Òàê êàê ýòî âåðíî äëÿíî ýòî çíà÷èò, ÷òî äëÿ òàêèõ n ω 6∈ Cnk ; òåì áîëåå ω 6∈n=1êàæäîãî ω ∈ Ω, òî ïîëó÷àåòñÿ, ÷òî∞\Cnk = ∅.n=1Âñïîìíèì êðèòåðèé ñ÷åòíîé àääèòèâíîñòè. Åñëè èìååò ìåñòî ïîñëåäîâàòåëüíîñòü èçìåðèìûõ ìíîæåñòâ∞\C1 ⊃ C2 ⊃ . . .

èCj = ∅, òî ν(Cj ) −→ 0.j=1Äîêàæåì òåïåðü, ÷òî ∀ ε > 0, ∀kZZlim (gn (ω) + ε) dν > ϕk (ω) dν.n→∞ΩΩÇàìåòèì, ÷òî ∀k, n¢¢¢R¡RR¡R¡Rgn (ω) + ε dν + ϕk (ω) dν =gn (ω) + ε dν +gn (ω) + ε dν + ϕk (ω) dν >Ω>RkCnRϕk (ω) dν +AknAknϕk (ω) dν +kCnÏðîâåðèì, ÷òî ∀ kRR¡kCn¢gn (ω) + ε dν =Rϕk (ω) dν +ΩkCnR¡kCn¢gn (ω) + ε dν.(1)kCnϕk (ω) dν → 0 ïðè n → ∞.kCnÏóñòü Mk = max ϕk (ω); òîãäàωZ∀ k, ∀ωϕk (ω) 6 Mk ⇒ZkCnòàê êàêZ∀ k, nZϕk (ω) dν >kCn¡Mk dν 6 Mk ν(Cnk ) −→ 0;ϕk (ω) dν 6kCnZ¢n→∞gn (ω) + ε dν, òî ∀ kkCn(gn (ω) + ε) dν −→ 0.kCnÏåðåõîäÿ ê ïðåäåëó (ïî n) â íåðàâåíñòâå (1) ïîëó÷èì:ZZZ¡¢∀klimgn (ω) + ε dν = limgn (ω) dν + εν(Ω) > ϕk (ω) dν.n→∞n→∞ΩΩΩÒàê êàê ýòî âåðíî äëÿ êàæäîãî ε > 0, òîZ∀klimgn (ω) dν > ϕk (ω) dν.n→∞Ω¤93 Ëåêöèÿ3 ËåêöèÿÑâîéñòâà èíòåãðàëà Ëåáåãà äëÿ ïðîèçâîëüíûõ f : Ω → R òàêèå æå, êàê è äëÿ ïðîñòûõíåîòðèöàòåëüíûõ ôóíêöèé:RRRF(1) Åñëè Ω = A1 A2 , òî f (ω) dν = f (ω) dν + f (ω) dν .ΩA1A2RR(2) cf (ω) dν = c f (ω) dν - âåðíî ∀c ∈ R.ΩΩRRR(3) (f1 (ω) + f2 (ω)) dν = f1 (ω) dν + f2 (ω) dν.ΩΩΩÄîêàçàòåëüñòâî.

Äîêàæåì ñâîéñòâî (3) èíòåãðàëà Ëåáåãà. Ðàçîáüåì Ω íà 6 ìíîæåñòâΩ=6GAj , ãäåj=1A1A2A3A4A5A6= {ω= {ω= {ω= {ω= {ω= {ω∈Ω∈Ω∈Ω∈Ω∈Ω∈Ω::::::f1 (ω) > 0, f2 (ω) > 0}f1 (ω) > 0, f2 (ω) < 0, f1 (ω) + f2 (ω) > 0}f1 (ω) > 0, f2 (ω) < 0, f1 (ω) + f2 (ω) < 0}f1 (ω) < 0, f2 (ω) > 0, f1 (ω) + f2 (ω) > 0}f1 (ω) < 0, f2 (ω) > 0, f1 (ω) + f2 (ω) < 0}f1 (ω) < 0, f2 (ω) < 0}6RRPg dν. Ñëåäîâàòåëüíî, äëÿ äîêàçàòåëüñòâà (3) äîÈç (1) âûòåêàåò ðàâåíñòâî g dν =j=1 AjΩñòàòî÷íî äîêàçàòü òàêîå æå ðàâåíñòâî äëÿ êàæäîãî Aj .Äëÿ A1 , A6 - î÷åâèäíî.

Ðàññìîòðèì òåïåðü ìíîæåñòâî A2 . Íóæíî äîêàçàòü, ÷òîZZZ(f1 + f2 ) dν = f1 dν + f2 dν.A2Çàïèøåì òîæäåñòâîA2A2¢¡¢ ¡f1 (ω) = f1 (ω) + f2 (ω) + −f2 (ω) ⇒{z} | {z }|>0>0ZZ¢¡f1 + f2 dν +f1 dν =A2ZZ(−f2 ) dν = −A2çíà÷èò,Zf2 dν,A2Zf1 dν +A2(−f2 ) dν.A2A2Èç ñâîéñòâà (2) ñëåäóåò, ÷òîZZf2 dν =A2(f1 + f2 ) dν.A2103 ËåêöèÿÄëÿ ìíîæåñòâ A3 , A4 , A5 äîêàçàòåëüñòâî àíàëîãè÷íî.¤Ïóñòü L̄1 (Ω, A, ν) - (ëèíåéíîå) ïðîñòðàíñòâî âñåõ èçìåðèìûõ è èíòåãðèðóåìûõ ïî ìåðåν ôóíêöèé. Èç îïðåäåëåíèÿ ñëåäóåò, ÷òî f ∈ L̄1 ⇔ | f |∈ L̄1 .Îáîçíà÷èì ÷åðåç L̄0 (Ω, A, ν) - ìíîæåñòâî âñåõ èçìåðèìûõ ôóíêöèé. Ãîâîðÿò, ÷òî èçìåðèìûå ôóíêöèè f, g ñîâïàäàþò ïî÷òè âñþäó, åñëè ìíîæåñòâîA = {ω : f (ω) 6= g(ω)} - èìååò ìåðó íóëü.Ïðîâåðèì, ÷òîZνA = 0 ⇒f dν = 0.AÄëÿ ïðîñòûõ íåîòðèöàòåëüíûõ ôóíêöèè ýòî ñëåäóåò ïðÿìî èç îïðåäåëåíèÿ; äëÿ íåîòðèöàòåëüíûõ èçìåðèìûõ ïîëó÷àåòñÿ ñ ïîìîùüþ ïåðåõîäà; äëÿ ïðîèçâîëüíûõ èçìåðèìûõ- èç ðàçëîæåíèÿ f = f + − f − .RRÓÒÂÅÐÆÄÅÍÈÅ 1.

Ïóñòü f = g ν ïî÷òè âñþäó, òîãäà f (ω) dν = g dν.ΩΩÄîêàçàòåëüñòâî. Ïóñòü A = {ω : f (ω) = g(ω)}. ÒîãäàZZZZZZZZf (ω) dν =f dν + f dν =f dν =g dν =g dν + g dν = g dν.ΩΩ\AAΩ\AΩ\AΩ\AAΩ¤Ðàññìîòðèì ïîäïðîñòðàíñòâî E0 = {f ∈ L̄0 (Ω, A, ν); f = 0 ν ïî÷òè âñþäó} â L̄0 (Ω, A, ν).Îïðåäåëåíèå 10. Äâå ôóíêöèè íàçûâàþòñÿ ýêâèâàëåíòíûìè, åñëè îíè ñîâïàäàþò ïî÷òè âñþäó.Óäîáíî ðàññìàòðèâàòü ôàêòîð-ïðîñòðàíñòâàL0 (Ω, A, ν) = L̄0 (Ω, A, ν)/E0 .L1 (Ω, A, ν) = L̄1 (Ω, A, ν)/E0 .Ýëåìåíò ýòîãî ôàêòîð-ïðîñòðàíñòâà - ýòî êëàññ ýêâèâàëåíòíîñòè. Áóäåì îáîçíà÷àòü ýëåìåíòû ïðîñòðàíñòâà L̄0 òàê æå, êàê è ýëåìåíòû ïðîñòðàíñòâà L0 , òî åñòü ïèñàòü f1 (·) ∈ L0 ,èìåÿ ââèäó, ÷òî f1 (·) - ýòî ïðåäñòàâèòåëü êëàññà ýêâèâàëåíòíîñòè.3.1 Íåðàâåíñòâî ×åáûøåâà.Ïóñòü f ∈ L1 (Ω, A, ν).

Òîãäà1∀ c > 0 ν{ω ∈ Ω : |f (ω)| > c} 6cZ|f (ω)| dν.ΩÄîêàçàòåëüñòâî. Ââåäåì äâà ìíîæåñòâàC1 = {ω : |f (ω)| > c}113 ËåêöèÿC2 = {ω : |f (ω)| < c}ÒîãäàRRRRRR|f (ω)| dν = |f (ω)| dν + |f (ω)| dν > |f (ω)| dν > c dν = c dν = c ν{ω : |f (ω)| > c}ΩC1C2C1C1C1¤ÓÒÂÅÐÆÄÅÍÈÅ 2. Åñëè f ∈ L1 (Ω, A, ν) , òî |f (ω)| < ∞ ν ïî÷òè âñþäó.Äîêàçàòåëüñòâî.∞\{ω : |f (ω)| = ∞} ={ω : |f (ω)| > n}.n=1Îöåíèì ìåðó êàæäîãî èç ýòèõ ìíîæåñòâ1ν{ω : |f (ω)| > n} 6nZ|f (ω)| dν −→ 0 (n → ∞).ΩÒàê êàêR|f (ω)| dν - ýòî êîíå÷íîå ôèêñèðîâàííîå ÷èñëî.Ωßñíî, ÷òî{ω : |f (ω)| > n} ⊃ {ω : |f (ω)| > n + 1}.Äàëåå ïîëüçóåìñÿ òåì, ÷òî äëÿ ïîñëåäîâàòåëüíîñòè âëîæåííûõ ìíîæåñòâ ìåðà èõ ïåðåñå÷åíèÿ ðàâíà ïðåäåëó ìåð ýòèõ ìíîæåñòâ.¤Ò Å Î Ð Å Ì À 2 (Áåïïî-Ëåâè).

Ïóñòü fn (ω) - ïîñëåäîâàòåëüíîñòü èçìåðèìûõ íåîòðèöàòåëüíûõ ôóíêöèé, òàêàÿ ÷òî fn (ω) % f (ω) â êàæäîé òî÷êå. ÒîãäàZZlimfn (ω) dν = f (ω) dνn→∞ΩΩ(â ÷àñòíîñòè, åñëè f - èíòåãðèðóåìà, òî ïðåäåë êîíå÷åí). (Åñëè â ôîðìóëèðîâêå äîáàâèòü ñëîâà ïî÷òè âñþäó, òî òåîðåìà âñå åùå áóäåò âåðíà).Äîêàçàòåëüñòâî. Ïóñòü gij - ïîñëåäîâàòåëüíîñòè ïðîñòûõ ôóíêöèé, òàêèå ÷òî0 6 g11 6 g12 . . . → f10 6 g21 6 g22 . .

. → f2...0 6 gk1 6 gk2 . . . → fkf1 6 f2 6 . . . 6 fk → f (ω)Ââåäåì íîâûå ôóíêöèè gn (ω) = max gkn (ω).Çàìåòèì, ÷òî ∀ ω16k6ng1 (ω) 6 g2 (ω) 6 . . . 6 gn (ω) 6 fn (ω) → f (ω).123 ËåêöèÿÑëåäîâàòåëüíî, â ñèëó ìîíîòîííîñòè ∃ lim gn (ω) = g(ω) è g(ω) 6 f (ω). Ïðåäïîëîæèì,n→∞÷òî äîêàçàíî, ÷òî ∀ ω : g(ω) = f (ω). Òîãäà ïî îïðåäåëåíèþ èíòåãðàëà ïîëó÷èì:ZZgn dν → f (ω) dνΩΩÒàê êàê∀n gn (ω) 6 fn (ω) 6 f (ω),òîZ∀nZgn (ω) dν 6ΩZfn (ω) dν 6Ωf (ω) dν.ΩÏåðåõîäÿ ê ïðåäåëó, ïîëó÷èìZZZf (ω) dν 6 lim fn (ω) dν 6 f (ω) dν.Òàêèì îáðàçîì,RΩfn dν →ΩRΩΩf (ω) dν .ΩÒåì ñàìûì, çàêëþ÷åíèå òåîðåìû äîêàçàíî.

Îñòàëîñü ïðîâåðèòü, ÷òî g(ω) = f (ω).Íàì èçâåñòíî, ÷òîn→∞∀k 6 n gkn (ω) 6 gn (ω) → g(ω) 6 f (ω).Çíà÷èò,∀k 6 n gkn (ω) 6 g(ω) 6 f (ω).Íî äëÿk→∞∀kgkn (ω) → fk (ω) 6 g(ω).Çíà÷èò,lim fk (ω) = f (ω) 6 g(ω) 6 f (ω) =⇒ f (ω) = g(ω).k→∞¤Çàìå÷àíèå.  òåîðåìå ìîæíî âìåñòî óñëîâèå 0 6 fn ìîæíî çàìåíèòü íà óñëîâèå F 6 f1 ,ãäå F - èíòåãðèðóåìàÿ ôóíêöèÿ. ×òîáû ñâåñòè ýòî ê ïðåäûäóùåìó ñëó÷àþ, äîñòàòî÷íîðàññìîòðåòü ïîñëåäîâàòåëüíîñòü0 6 f1 (ω) − F (ω) 6 f2 (ω) − F (ω) 6 . . .Ò Å Î Ð Å Ì À 3 (Ôàòó-Ëåáåãà). Ïóñòü (fn ) - ïîñëåäîâàòåëüíîñòü èçìåðèìûõ ôóíê-öèé, ïðè÷åì∀ n, ∀ ωÒîãäàfn (ω) > F (ω), ãäå F ∈ L1 (Ω, A, ν).ZZfn dνlim inf fn (ω) dν 6 lim infn→∞è åñëèn→∞ΩΩZ∀n fn (ω) 6 F (ω), òîZlim fn (ω) dν > limfn (ω) dν.n→∞n→∞ΩΩ134 ËåêöèÿÄîêàçàòåëüñòâî.Íàïîìíèì, ÷òîdeflim inf fn (ω) = lim {inf fk (ω)} = lim fn (ω).n→∞n→∞ k>nn→∞Ââåäåì ïîñëåäîâàòåëüíîñòü ôóíêöèégn (ω) = {inf fk (ω)} ⇒ ∀k > n gn (ω) ≤ gk (ω).k>nÌû çíàåì, ÷òîF (ω) ≤ g1 (ω) ≤ g2 (ω) ≤ .

. . → lim fn (ω).n→∞RRÑëåäîâàòåëüíî ïî òåîðåìå Áåïïî-Ëåâèlim gn dν = lim fn (ω) dν.n→∞ ΩΩ n→∞ZZZZ∀k > ngn (ω) dν 6 fk (ω) dν ⇒ ∀ngn (ω) dν 6 limfk (ω) dν ⇒ΩΩZ⇒ limZgn (ω) dν 6 limn→∞k→∞ΩZk→∞ΩZlim fn (ω) dν 6 limΩn→∞n→∞Ωfk (ω) dν.Ωfn (ω) dν.ΩÒàêèì îáðàçîì, äîêàçàíà ïåðâàÿ ÷àñòü òåîðåìû. Äîêàæåì òåïåðü âòîðóþ ÷àñòü.−F (ω) 6 −f (ω),¡¢lim −fn (ω) = − lim fn (ω),n→∞n→∞ZZZZ¡¢¡¢¡¢−lim fn (ω) dν =lim −fn (ω) dν 6 lim − fn (ω) dν = − limfn (ω) dν,n→∞ΩΩn→∞k→∞Zòàê ÷òîn→∞ΩΩZlim fn (ω) dν > limn→∞fn (ω) dν.n→∞ΩΩ¤Çàìå÷àíèå.  òåîðåìå Ôàòó-Ëåáåãà ìîæíî ïðåäïîëàãàòü, ÷òî âñå åå óñëîâèÿ âûïîëíåíûëèøü ïî÷òè âñþäó.4 ËåêöèÿÒ Å Î Ð Å Ì À 4 (Ëåáåãà î ìàæîðèðóåìîé ñõîäèìîñòè). (Î ïðåäåëüíîì ïåðåõîäå ïîäçíàêîì èíòåãðàëà.) Ïóñòü F ∈ L1 (Ω, A, ν) è (fn ) - ïîñëåäîâàòåëüíîñòü èçìåðèìûõ ôóíêöèé, òàêàÿ ÷òî fn (ω) → f (ω) ν -ïî÷òè âñþäó.