Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 30 стр.

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МГУ | Москва

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{f
n
}
∀n : f
n+1
(x) ≥ f
n
(x) ,
sup{I
0
(f
n
) | 1 ≤ n < ∞} = C < ∞.
∃f(x) : f(x) = lim
n→∞
f
n
(x) < ∞.
∀n : mes{x | f
n
(x) > f
n+1
(x)} = 0.
mes{x | lim
n→∞
f
n
(x) = ∞} = 0.
g
n
(x) := max{(f
k
(x) − f
1
(x))
+
| k ≤ n}.
Z := {x | lim
n→∞
f
n
(x) = ∞} = {x | lim
n→∞
g
n
(x) = ∞}
g
n
(x)
∀(x , n) : g
n+1
(x) ≥ g
n
(x) , g
n
(x) = f
n
(x) − f
1
(x),
I
0
(g
n
) = I
0
(f
n
) − I
0
(f
1
)
∀n : I
0
(g
n
) ≤ 2C.
g
n
(x)/2C
∀n : I
0
(g
n
(x)/2C) < , ∀(x ∈ Z) : sup{g
n
(x)/2C | 1 ≤ n < ∞} = ∞.
Ëåììà 1.1.4. Ïóñòü ïîñëåäîâàòåëüíîñòü ýëåìåíòàðíûõ ôóíêöèé {f }                 n
óäîâëåòâîðÿåò äâóì óñëîâèÿì:

                        ∀n : ï.â. fn+1 (x) ≥ fn (x) ,                        (1.24)
                        sup{I0 (fn ) | 1 ≤ n < ∞} = C < ∞ .                  (1.25)

Òîãäà
                       ï.â.    ∃f (x) : f (x) = lim fn (x) < ∞.              (1.26)
                                                   n→∞

   Äîêàçàòåëüñòâî. Âî-ïåðâûõ çàìåòèì, ÷òî óñëîâèå (1.24) ýêâèâàëåíòíî
óñëîâèþ:
                  ∀n : mes{x | fn (x) > fn+1 (x)} = 0.          (1.27)
Âî-âòîðûõ çàìåòèì, ÷òî óòâåðæäåíèå ëåììû ýêâèâàëåíòíî óòâåðæäå-
íèþ:
                   mes{x | lim fn (x) = ∞} = 0.           (1.28)
                                       n→∞

Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü

                     gn (x) := max{(fk (x) − f1 (x))+ | k ≤ n}.

ßñíî, ÷òî

               Z := {x | lim fn (x) = ∞} = {x | lim gn (x) = ∞}              (1.29)
                           n→∞                           n→∞

Ïîñëåäîâàòåëüíîñòü íåîòðèöàòåëüíûõ ýëåìåíòàðíûõ ôóíêöèé gn (x) óäî-
âëåòâîðÿåò óñëîâèÿì:

             ∀(x , n) : gn+1 (x) ≥ gn (x) , ï.â. gn (x) = fn (x) − f1 (x),

ïîýòîìó

             I0 (gn ) = I0 (fn ) − I0 (f1 )

è

             ∀n : I0 (gn ) ≤ 2C.

Ñëåäîâàòåëüíî, ìîíîòîííî íåóáûâàþùàÿ ïîñëåäîâàòåëüíîñòü íåîòðèöà-
òåëüíûõ ýëåìåíòàðíûõ ôóíêöèé gn (x)/2C óäîâëåòâîðÿåò óñëîâèÿì (1.22)
äëÿ ìíîæåñòâà (1.29), òàê êàê

    ∀n : I0 (gn (x)/2C) <  , è ∀(x ∈ Z) : sup{gn (x)/2C | 1 ≤ n < ∞} = ∞.

Ëåììà äîêàçàíà.

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