Space of Integrable Functions is Vector Space
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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): $\S 10$