This is the definition from Terry Tao's Introduction to measure theory:
A property $P(x)$ of a point $x \in \mathbb{R}^d$ is said to hold (Lebesgue) almost everywhere in $\mathbb{R}^d$ or for (Lebesgue) almost every point $x \in \mathbb{R}^d$, if the set of $x \in \mathbb{R}^d$ for which $P(x)$ fails has Lebesgue measure zero (i.e. $P$ is true outside of a null set).
My question is that what if the set of $x \in \mathbb{R}^d$ for which $P(x)$ fails is not Lebesgue measurable? Then can we say whether $P(x)$ is true almost everywhere or not?