Minkowski's Inequality/Lebesgue Spaces

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

Let $p \in \closedint 1 \infty$.

Let $f, g: X \to \R$ be $p$-integrable, that is, elements of Lebesgue $p$-space $\map {\LL^p} \mu$.

Then their pointwise sum $f + g: X \to \R$ is also $p$-integrable, and:


 * $\norm {f + g}_p \le \norm f_p + \norm g_p$

where $\norm {\, \cdot \, }_p$ denotes the $p$-seminorm.

Proof
We split into three cases.

Case 1: $p > 1$
We first show that $f + g \in \map {\LL^p} \mu$.

Note that from Pointwise Maximum of Measurable Functions is Measurable:


 * $x \mapsto \max \set {\map f x, \map g x}$ is $\Sigma$-measurable.

We then have from Measure is Monotone:


 * $\ds \int \size {f + g}^p \rd \mu = \int \size {2 \max \set {\map f x, \map g x} }^p \map {\rd \mu} x$

We then have:

Since $f, g \in \map {\LL^p} \mu$, we have:


 * $\ds \int \size f^p \rd \mu < \infty$

and:


 * $\ds \int \size g^p \rd \mu < \infty$

so:


 * $\ds \int \size {f + g}^p \rd \mu < \infty$

so:


 * $f + g \in \map {\LL^p} \mu$

If:


 * $\ds \int \size {f + g}^p \rd \mu = 0$

then the desired inequality is immediate.

So, take:


 * $\ds \int \size {f + g}^p \rd \mu > 0$

Write:


 * $\ds \int \size {f + g}^p \rd \mu = \int \size {f + g} \size {f + g}^{p - 1} \rd \mu$

From the Triangle Inequality, Integral of Positive Measurable Function is Monotone and Integral of Positive Measurable Function is Additive, we have:


 * $\ds \int \size {f + g} \size {f + g}^{p - 1} \rd \mu \le \int \size f \size {f + g}^{p - 1} \rd \mu + \int \size g \size {f + g}^{p - 1} \rd \mu$

From Hölder's Inequality for Integrals, we have:


 * $\ds \int \size f \size {f + g}^{p - 1} \rd \mu + \int \size g \size {f + g}^{p - 1} \rd \mu \le \paren {\int {\size f}^p \rd \mu}^{1/p} \paren {\int \size {f + g}^{q \paren {p - 1} } \rd \mu}^{1/q} + \paren {\int {\size g}^p \rd \mu}^{1/p} \paren {\int \size {f + g}^{q \paren {p - 1} } \rd \mu}^{1/q}$

where $q$ satisfies:


 * $\ds \frac 1 p + \frac 1 q = 1$

Then we have:


 * $p + q = p q$

so:


 * $p = pq - q = q \paren {p - 1}$

So we have:


 * $\ds \int \size {f + g}^p \rd \mu \le \paren {\paren {\int {\size f}^p \rd \mu}^{1/p} + \paren {\int {\size g}^p \rd \mu}^{1/p} } \paren {\int \size {f + g}^p \rd \mu}^{1/q}$

From the definition of the $p$-seminorm we have:


 * $\ds \int \size {f + g}^p \rd \mu \le \paren {\norm f_p + \norm g_p} \paren {\int \size {f + g}^p \rd \mu}^{1/q}$

So that:


 * $\ds \paren {\int \size {f + g}^p \rd \mu}^{1 - 1/q} \le \norm f_p + \norm g_p$

That is:


 * $\ds \paren {\int \size {f + g}^p \rd \mu}^{1/p} \le \norm f_p + \norm g_p$

So from the definition of the $p$-seminorm we have:


 * $\norm {f + g}_p \le \norm f_p + \norm g_p$

Case 2: $p = 1$
From the Triangle Inequality, we have:


 * $\size {f + g} \le \size f + \size g$

So, from Integral of Positive Measurable Function is Additive and Integral of Positive Measurable Function is Monotone, we have:


 * $\ds \int \size {f + g} \rd \mu \le \int \size f \rd \mu + \int \size g \rd \mu$

So if $f, g \in \map {\LL^1} \mu$ we have $f + g \in \map {\LL^1} \mu$

From the definition of the $1$-seminorm, we also have that:


 * $\norm {f + g}_1 \le \norm f_1 + \norm g_1$

immediately.

Case 3: $p = \infty$
Suppose $f, g \in \map {\LL^\infty} \mu$.

Then from the definition of the $\LL^\infty$-space, there exists $\mu$-null sets $N_1$ and $N_2$ such that:


 * $\size {\map f x} \le \norm f_\infty$ for $x \not \in N_1$

and:


 * $\size {\map g x} \le \norm g_\infty$ for $x \not \in N_2$

Then, for $x \not \in N_1 \cup N_2$ we have:


 * $\size {\map f x + \map g x} \le \norm f_\infty + \norm g_\infty$

by the Triangle Inequality.

From Null Sets Closed under Countable Union, we have:


 * $N_1 \cup N_2$ is $\mu$-null.

So:


 * $\norm {f + g}_\infty \le \norm f_\infty + \norm g_\infty$

as desired.