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 by definition of p-integrable function:
:$f + g \in \map {\LL^p} \mu$ 
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.
== Sources ==
* : $6.12$: The 'elementary formula' for expectation
* : $12.4$
Category:Minkowski's Inequality
Category:Lebesgue Spaces