Definition:Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality

Definition

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

Let $\map \MM {X, \Sigma, \R}/\sim_\mu$ be the space of real-valued measurable functions identified by $\mu$-A.E. equality.

We define pointwise addition $+$ on $\map \MM {X, \Sigma, \R} / \sim_\mu$ by:

$\eqclass f {\sim_\mu} + \eqclass g {\sim_\mu} = \eqclass {f + g} {\sim_\mu}$

where:

$\eqclass f {\sim_\mu}, \eqclass g {\sim_\mu} \in \map \MM {X, \Sigma, \R} / \sim_\mu$

$f + g$ denotes the usual pointwise sum of $f$ and $g$.

Also see

• Pointwise Addition on Space of Real-Valued Measurable Functions Identified by A.E. Equality is Well-Defined