Definition:Space of Measurable Functions Identified by A.E. Equality/Real-Valued Function/Vector Space
Jump to navigation
Jump to search
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:
- $\struct {\map \MM {X, \Sigma, \R} / \sim_\mu, +, \cdot}_\R$