Definition:Space of Measurable Functions Identified by A.E. Equality/Real-Valued Function/Vector Space

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

Let $\map \MM {X, \Sigma, \R}$ be the set of real-valued $\Sigma$-measurable functions on $X$.

Let $\sim_\mu$ be the almost-everywhere equality relation on $\map \MM {X, \Sigma, \R}$ with respect to $\mu$. Let $+$ denote pointwise addition on $\map \MM {X, \Sigma, \R}/\sim_\mu$.

Let $\cdot$ be pointwise scalar multiplication on $\map \MM {X, \Sigma, \R}/\sim_\mu$.

Then we define the vector space $\map \MM {X, \Sigma, \R}/\sim_\mu$ as:


 * $\struct {\map \MM {X, \Sigma, \R} / \sim_\mu, +, \cdot}_\R$

Also see

 * Space of Measurable Functions Identified by A.E. Equality is Vector Space