Space of Integrable Functions is Vector Space

Theorem

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

Let $\map {\LL^1} \mu$ be the space of real-valued $\mu$-integrable functions.


Then $\map {\LL^1} \mu$, endowed with pointwise $\R$-scalar multiplication and pointwise addition, forms a vector space over $\R$.


Proof


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Sources

  • 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 10$
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