Minkowski's Inequality/Lebesgue Spaces

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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:

\(\ds \int \size {2 \max \set {\map f x, \map g x} }^p \map {\rd \mu} x\) \(=\) \(\ds \int 2^p \size {\max \set {\map f x, \map g x} }^p \map {\rd \mu} x\) Integral of Positive Measurable Function is Positive Homogeneous
\(\ds \) \(=\) \(\ds 2^p \int \max \set {\size {\map f x}^p, \size {\map g x}^p} \map {\rd \mu} x\)
\(\ds \) \(\le\) \(\ds 2^p \int \paren {\size f^p + \size g^p} \rd \mu\)

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:

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

$\Box$


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.

$\Box$


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.

$\blacksquare$


Source of Name

This entry was named for Hermann Minkowski.


Sources

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