- #1

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*Элементы теории функций и функционального анализа,*of Lebesgue integral of measurable function ##f:X\to \mathbb{C}## on ##X,\mu(X)<\infty## as the limit

##\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})##

where ##\{f_n\}## is a sequence of simple, i.e. taking

**countably many**(not necessarily finitely) values ##y_{n,k}## for ##k=1,2,\ldots##, functions ##f_n:X\to\mathbb{C}##

**uniformly**converging to ##f##, and ##\{y_{n,k}\}=f_n(A_{n,k})## where ##\forall i\ne j\quad A_{n,i}\cap A_{n,j}=\emptyset##. If ##\mu## is not finite but is ##\sigma##-finite with ##X=\bigsqcup_{j=1}^\infty X_j## then ##\int_{X}fd\mu:=\sum_{j=1}^\infty\int_{X_j}fd\mu##, provided that ##\sum_{j=1}^\infty\int_{X_j}|f|d\mu<\infty##.

I read other authors defining the Lebesgue integral for non-negative functions ##g:X\to[0,+\infty]##, where ##\mu(X)\le\infty##, by using simple non-negative functions ##s_i## taking only

**finitely many**values on measurable sets ##A_i## in the following way:

##\int_X gd\mu:=\sup\{\sum_i s_i\mu(A_i):\forall x\in X\quad s_i(x)\le g(x)\}##

and in general in the following way, provided that ##\int_X|f|d\mu<\infty##:

##\int_Xfd\mu:=\int_X \text{Re}f_+d\mu-\int_X \text{Re}f_-d\mu+i(\int_X \text{Im}f_+d\mu-\int_X \text{Im}f_-d\mu)##

where subscript ##\pm## is used to define ##g_+(x):=\max\{g(x),0\}## and ##g_-(x):=-\min\{g(x),0\}## for any measurable function ##g##.

I have not been able to find more about it, since the texts that are available to me all use Kolmogorov-Fomin's definition, but I am convinced that such definitions are equivalent. From what I know -I have finished Kolmogorov-Fomin's

*Элементы теории функций и функционального анализа*, which significantly overlaps with the English language translation

*Introductory Real Analysis*by the same authors- I would say that it would be enough to prove the equivalence for non-negative functions ##g:X\to[0,+\infty)##.

If they are equivalent, how can it be proved?

I uncountably thank you and wish you a happy 2015!