## Lebesgue measure

Jordan measure: finite union of element sets (boxes)

Lebesgue measure: conutable union of element sets (boxes)

## Properties of Lebesgue measure

- Monotonicity

- Subadditivity

## Simple function

- Finite linear combination of measurable set

- \(C_1\)\(1_{E_1}\) \(+\) \(C_2\)\(1_{E_2}\) \(...\) \(+\) \(C_k\)\(1_{E_k}\) where \(1_{E_k}\) is identity function

## Uniform convergence

Sequence of function \(f_n\)\((x)\) is uniform convergence, if \(lim_{n\rightarrow \inf}\) \(Sup\)

\(Sup\) is the point in uniform convergence