# Lebesgue measure

## Lebesgue measure

Jordan measure: finite union of element sets (boxes)
Lebesgue measure: conutable union of element sets (boxes)

## Properties of Lebesgue measure

• Monotonicity
• $$C_1$$$$1_{E_1}$$ $$+$$ $$C_2$$$$1_{E_2}$$ $$...$$ $$+$$ $$C_k$$$$1_{E_k}$$ where $$1_{E_k}$$ is identity function
Sequence of function $$f_n$$$$(x)$$ is uniform convergence, if $$lim_{n\rightarrow \inf}$$ $$Sup$$
$$Sup$$ is the point in uniform convergence