Jtdylktuy

Consider a real function $f$ and take $mathbb R$ together with Lebesgue measure $m$. I want to check if I have the right reasoning with regards to working with the following "almost everywhere" statement; I have that $f(x)neq0$ for almost all $xinmathbb R$.

Now, I know that this means that the set of $xinmathbb R$ for which $f(x)=0$ is a null set with respect to Lebesgue measure. That is, $m({xinmathbb R:f(x)=0})=0$. But since the statement that $f(x)neq0$ holds for almost all $xinmathbb R$, does that mean that $m({xinmathbb R:f(x)neq0})=m(mathbb R)=+infty$? My thinking is that since the statement holds for all $mathbb R$ except possibly on a subset of $mathbb R$ with measure zero, then the measure of the set on which the statement holds must coincide with the measure of $mathbb R$; is this the correct reasoning?

No issues with what you have said : $f(x) neq 0$ almost everywhere, means that the set ${x : f(x) = 0}$ has Lebesgue measure zero, which implies that the set ${x : f(x) neq 0}$ has the same Lebesgue measure as $mathbb R$, which is $+infty$.

However, note that the converse is not true : if I tell you that the set ${x : f(x) neq 0}$ has measure $+infty$, then this does not imply that $f(x) neq 0$ almost everywhere. For example, take $f(x) = 0$ if $[x]$ is a multiple of $2$, and $1$ otherwise. Then ${x : f(x) neq 0}$ and ${x : f(x) = 0}$ have measure $+infty$.

In short, I want to tell you that when you say "$f(x) neq 0$ almost everywhere means ${x : f(x) neq 0}$ has measure infinite", we say it with the idea that the left side of the "means" is like a definition and the right side is an equivalent statement. However, the "means" for you, is an implication : the statements are not equivalent, as I showed. You should therefore be careful about a definition and an implication : that ${f(x) =0}$ has Lebesgue measure zero is the definition of $f(x) neq 0$ almmost everywhere. On the other hand, that ${f(x) neq 0}$ has full measure is an implication of $f(x) neq 0$ everywhere, and not equivalent to it. As long as you are aware of this, you can be relatively be relaxed about what you are writing.