Scalar Multiple of Integrable Function is Integrable Function

Theorem

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

Let $f: X \to \overline \R$ be a $\mu$-integrable function, and let $\lambda \in \R$.

Then $\lambda f: X \to \overline \R$, the pointwise $\lambda$-multiple of $f$, is also $\mu$-integrable.

That is, the space of integrable functions $\LL^1_{\overline \R}$ is closed under pointwise $\R$-scalar multiplication.

Proof

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.4 \, \text{(i)}$