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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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