Lebesgue Space is Vector Space

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Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space and let $p \in \closedint 1 \infty$.

Let $\map {\LL^p} {X, \Sigma, \mu}$ be Lebesgue $p$-space on $\struct {X, \Sigma, \mu}$.


Then $\map {\LL^p} {X, \Sigma, \mu}$ is a vector subspace of $\map \MM {X, \Sigma, \R}$, the space of real-valued $\Sigma$-measurable functions on $X$.

In particular, it is a vector space over $\R$.


Proof

From Space of Real-Valued Measurable Functions is Vector Space:

$\map \MM {X, \Sigma, \R}$ forms a vector space with pointwise addition and pointwise scalar multiplication.

From construction, we have:

$\map {\LL^p} {X, \Sigma, \mu} \subseteq \map \MM {X, \Sigma, \R}$

Since $0 \in \map {\LL^p} {X, \Sigma, \mu}$, we have $\map {\LL^p} {X, \Sigma, \mu} \ne \O$.

From the One-Step Vector Subspace Test, it suffices to show that for each $f, g \in \map {\LL^p} {X, \Sigma, \mu}$ and $\lambda \in \R$ we have:

$f + \lambda g \in \map {\LL^p} {X, \Sigma, \mu}$


Case 1: $1 \le p < \infty$

Since:

$\ds \paren {\int \size g^p \rd \mu}^{1/p} < \infty$

we have:

\(\ds \paren {\int \size {\lambda g}^p \rd \mu}^{1/p}\) \(=\) \(\ds \paren {\int \size \lambda^p \size g^p \rd \mu}^{1/p}\)
\(\ds \) \(=\) \(\ds \paren {\size \lambda^p \int \size g^p \rd \mu}^{1/p}\) Integral of Positive Measurable Function is Positive Homogeneous
\(\ds \) \(=\) \(\ds \size \lambda \paren {\int \size g^p \rd \mu}^{1/p}\)
\(\ds \) \(<\) \(\ds \infty\)

From Minkowski's Inequality on Lebesgue Space, we have:

$f + \lambda g \in \map {\LL^p} {X, \Sigma, \mu}$

as required.

$\Box$


Case 2: $p = \infty$

From $g \in \map {\LL^\infty} {X, \Sigma, \mu}$, we have:

$\map \mu {\set {x \in X : \size {\map g x} \ge c} } = 0$ for some $c > 0$.

Then we have:

$\map \mu {\set {x \in X : \size {\map g x} \ge c} } = \map \mu {\set {x \in X : \size {\lambda \map g x} \ge \size \lambda c} } = 0$

So $\lambda g \in \map {\LL^\infty} {X, \Sigma, \mu}$.

Then, from Minkowski's Inequality on Lebesgue Space, we have:

$f + \lambda g \in \map {\LL^\infty} {X, \Sigma, \mu}$

as required.

$\blacksquare$


Sources

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