I am struggling with convincing myself that if you equip [itex]\mathbb Z[/itex] with the counting measure [itex]m[/itex], the [itex]L^p[/itex] norm of measurable functions [itex]f: \mathbb Z \to \mathbb C[/itex] looks like(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\| f \|_p = \left( \sum_{n = -\infty}^\infty |a_n|^p \right)^{1/p}.

[/tex]

I know that any function on [itex]\mathbb Z[/itex] is essentially just a doubly-infinite sequence of complex numbers [itex]\left\{ \ldots, a_{-2}, a_{-1}, a_0, a_1, a_2, \ldots \right\}[/itex], with [itex]a_n = f(n)[/itex]. But how does one get from the general definition of integrals of positive functions (which [itex]|f|^p[/itex] certainly is) to the sum that appears above?

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# Integration with respect to the counting measure

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