Almost every

Properties

• If property ${\displaystyle P}$ holds almost everywhere and implies property ${\displaystyle Q}$, then property ${\displaystyle Q}$ holds almost everywhere. This follows from the monotonicity of measures.
• If ${\displaystyle (P_{n})}$ is a finite or a countable sequence of properties, each of which holds almost everywhere, then their conjunction ${\displaystyle \forall nP_{n}}$ holds almost everywhere. This follows from the countable sub-additivity of measures.
• By contrast, if ${\displaystyle (P_{x})_{x\in \mathbf {R} }}$ is an uncountable family of properties, each of which holds almost everywhere, then their conjunction ${\displaystyle \forall xP_{x}}$ does not necessarily hold almost everywhere. For example, if ${\displaystyle \mu }$ is Lebesgue measure on ${\displaystyle X=\mathbf {R} }$ and ${\displaystyle P_{x}}$ is the property of not being equal to ${\displaystyle x}$ (i.e. ${\displaystyle P_{x}(y)}$ is true if and only if ${\displaystyle y\neq x}$), then each ${\displaystyle P_{x}}$ holds almost everywhere, but the conjunction ${\displaystyle \forall xP_{x}}$ does not hold anywhere.