Triangle Inequality for Series/Lebesgue Spaces

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $\sequence {f_n}_{n \mathop \in \N} \in \map {\LL^p} {\mu}$ be a sequence of $p$-integrable functions, that is, a sequence in Lebesgue $p$-space.


Suppose that for all $n \in \N$, $f_n \ge 0$ holds pointwise.


This article is complete as far as it goes, but it could do with expansion.
In particular: This condition could possibly be weakened/altered, using Lebesgue's Dominated Convergence Theorem in place of Beppo Levi's Theorem
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Then:

$\ds \norm {\sum_{n \mathop = 1}^\infty f_n}_p \le \sum_{n \mathop = 1}^\infty \norm {f_n}_p$

where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.


Proof


This theorem requires a proof.
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Sources

  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.6$
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