Abstract
AbstractLet $\{f_{n}\}_{n \in \mathbb {N}}$
{
f
n
}
n
∈
N
be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$
(
Ω
,
F
,
μ
)
. Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$
lim
n
↑
∞
f
n
exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$
lim
n
↑
∞
∫
f
n
d
μ
=
∫
lim
n
↑
∞
f
n
d
μ
.
Funder
Japan Society for the Promotion of Science
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
Reference9 articles.
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