Measure theory, notation question, a.e.

Question

I have in my notes

$$\|f\|_\infty := \inf\{c\in \mathbb{R} : |f(\omega)|\leq c \quad a.e. \}$$

Can I take that to mean

$$\|f\|_\infty := \inf\{c\in \mathbb{R} : |f(\omega)|\leq c \quad \forall \omega \in A \; st\; \mu(A^c)=0 \}$$

Answer 1

No, your notation is incorrect. When you write something like $$\forall \omega \in A \; \text{such that}\; \mu(A^c)=0$$ you are quantifying over $\omega$, and the variable $A$ is free. So this expression only is meaningful if you have already given a definition of $A$, and you are making an assertion about all $\omega$ which are in this $A$ with the property that $\mu(A^c)=0$ (which is a queer thing to say, since the condition $\mu(A^c)=0$ does not involve $\omega$ at all).

Instead you could say something like $$\|f\|_\infty := \inf\{c\in \mathbb{R} : \exists A \text{ such that } \mu(A^c)=0 \text{ and } \forall\omega\in A, |f(\omega)|\leq c \}.$$

More simply, you could also just pick a specific $A$, since if any $A$ works, then $A=\{\omega:|f(\omega)|\leq c\}$ works. So you could just say $$\|f\|_\infty := \inf\{c\in \mathbb{R} : \mu(\{\omega:|f(\omega)|>c\})=0\}.$$