In Rudin's book there is the statement that an element of $ l^p $ may be regarded as a complex sequence of the form $ x = \{ {\xi}_i \}_{i \in \mathbb N} $, and $ ||x|| := {\{ \sum_{i=1}^{\infty} |{\xi}_i|^p \} }^{1/p} $.
Where we have a countable set $ A $, $ \sharp $ is the count measure and the measure space is $ (A, P(A), \sharp ) $.
I tried to prove it formally but I'm having difficulty. My difficulty is in WRITING the integral of an element of $ l^p $.
In the case that $ A $ is finite then there is a bijection $ A \approx \{1, ..., n \} $ and $ f \in l^p $ is a finite complex sequence of the form $ f = \{ {\xi}_i \}_{i = 1}^n $. But it is a simple function and so we can write in the form $ f = \sum_{i=1}^n f(i){\chi}_{\{i\}} $. Thus $ \int_{A} fd\sharp = \sum_{i=1}^n f(i)\sharp (\{i\}) = \sum_{i=1}^n {\xi}_{i} $ where $ f(i) = {\xi}_i \in \mathbb{C} $. Thus $ ||f|| := {\{ \sum_{i=1}^n |{\xi}_i|^p \} }^{1/p} $.
Now in the case that $ A $ is infinite then there is a bijection $ A \approx \mathbb N $ and $ f \in l^p $ is a complex sequence of the form $ f = \{ {\xi}_i \}_{i \in \mathbb N} $.
How can I justify that the integral of $ f $ is of the form $ \int_{A} fd\sharp = \sum_{i=1}^{\infty} f(i)\sharp (\{i\}) = \sum_{i=1}^{\infty} {\xi}_{i} $?