L-2 Inner Product is Well-Defined

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

Let $\map {\LL^2} {X, \Sigma, \mu}$ be the Lebesgue $2$-space of $\struct {X, \Sigma, \mu}$.

Let $\sim$ be the $\mu$-almost-everywhere equality relation on $\map {\LL^2} {X, \Sigma, \mu}$.

Let $\map {L^2} {X, \Sigma, \mu}$ be the $L^p$ space on $\struct {X, \Sigma, \mu}$.

Let $E, F \in \map {L^2} {X, \Sigma, \mu}$.

Then the $L^2$ inner product $\innerprod E F$ is well-defined.

Proof
Let $E = \eqclass f \sim$ and $F = \eqclass g \sim$ where $\sim$ is the $\mu$-almost everywhere equality relation.

We aim to show that:


 * $\ds \int \paren {f \cdot g} \rd \mu$

exists as a real number and is independent of the choice of representatives $f$ and $g$ of $E$ and $F$.

From the definition of the $L^2$ space, we have $f, g \in \map {\LL^2} {X, \Sigma, \mu}$.

From Hölder's Inequality, we have $f \cdot g \in \map {\LL^1} {X, \Sigma, \mu}$.

So the $\mu$-integral of $f \cdot g$ is well-defined and exists as a real number.

Now suppose that:


 * $\eqclass {f_1} \sim = \eqclass {f_2} \sim = E$

and:


 * $\eqclass {g_1} \sim = \eqclass {g_2} \sim = F$

From Equivalence Class Equivalent Statements, we have:


 * $f_1 \sim f_2$

and:


 * $g_1 \sim g_2$

That is:


 * $f_1 = f_2$ $\mu$-almost everywhere

and:


 * $g_1 = g_2$ $\mu$-almost everywhere.

From Pointwise Multiplication preserves A.E. Equality, we have:


 * $f_1 \cdot g_1 = f_2 \cdot g_2$ $\mu$-almost everywhere.

From A.E. Equal Positive Measurable Functions have Equal Integrals, we have:


 * $\ds \int \paren {f_1 \cdot g_1} \rd \mu = \int \paren {f_2 \cdot g_2} \rd \mu$

as desired.