Absolute Continuity

Definition

$\mu \ll \lambda$ Consider measures $\mu ,\lambda$ on a Measurable Space $(\Omega ,\mathcal{F})$. The measure $\mu$ said to be absolutely continuous with respect to $\lambda$, or $\mu$ is dominated by $\lambda$, denoted by $\mu \ll \lambda$, If $\forall S\in \mathcal{F},\lambda (S)=0\Rightarrow \mu (S)=0$

Summary

Comparison of conditions

Function	Derivative	Radon–Nikodym Derivative
Continuous Function	X	X (Cantor Function)
Absolute continuous function	X	O
Differentiable function	O	O

• Differentiable function $\subset$ Absolute continuous function $\subset$ Continuous function

Facts

Continuously differentiable $\subset$ Lipschitz continuous $\subset$ $\alpha -$Holder Continuous $\subset$ Uniformly Continuous $\subset$ Continuous Lipschitz continuous Absolute continuous $\subset$ Uniformly Continuous $\subset$ Continuous where $0<0\le 1$

Link to original