User:Caliburn/s/mt/Quotient Space of Measurable Functions forms Vector Space

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

Let $\map {\mathcal M} {X, \Sigma}$ be the space of $\Sigma$-measurable functions $f : X \to \R$.

Let $\sim$ be the equivalance on $\map {\mathcal M} {X, \Sigma}$ defined by:


 * $f \sim g$ $f = g$ $\mu$-almost everywhere.

Consider the quotient space $\map {\mathcal M} {X, E}/\sim$.

For each $\eqclass f \sim, \eqclass g \sim \in \map {\mathcal M} {X, E}/\sim$ and $\lambda \in \overline \R$ define:


 * $\eqclass f \sim +_{\map {\mathcal M} {X, E}/\sim} \eqclass g \sim = \eqclass {f + g} \sim$

and:


 * $\lambda \circ_{\map {\mathcal M} {X, E}/\sim} \eqclass f \sim = \eqclass {\lambda f} \sim$

Then $\struct {\map {\mathcal M} {X, E}/\sim, +_{\map {\mathcal M} {X, E}/\sim}, \circ_{\map {\mathcal M} {X, E}/\sim} }_\R$ forms an $\R$-vector space.