TITLE: Have I understood Compact Set correctly
QUESTION [1 upvotes]: In our current Measure Theory Class, we bought up the notion for a function $f:\mathbb R \to \mathbb R$ that is continuous to have a compact support, is equivalent to the fact that $\overline{\{x \in \mathbb R: f(x)\neq 0\}}$ needs to be compact. 
Note $C_{c}(\mathbb R)=\{ f: \mathbb R \to \mathbb R:f$ is continuous and $supp f$ compact $\}$
I assume the most important aspect to take out of the notion of compact support (at least for my class in Measure Theory) is the fact that: $\forall p \in [1,\infty]:$

$C_{c}(\mathbb R)\subseteq L^{p}$

So if I need to prove a function is $p-$integrable then I can simply show that it has compact support $supp f$.
Is this the most applicable case to use the notion of a compact support in Measure Theory?
Other Question:
Considering the factor $\partial \{x \in \mathbb R: f(x)\neq 0\}$, does this mean that the set $\overline{\{x \in \mathbb R: f(x)\neq 0\}}$ can contain points $y \in \mathbb R$ such $f(y)=0$ but these sets of points need to be countable? Or can they be uncountably infinite?

REPLY [1 votes]: In measure theory in isolation the concern is really more about finite measure support (where support has no closure operation in that setting, in part because there is no guarantee that a topology has been defined on the measure space). The point there is that a bounded measurable function with finite measure support is in all of the $L^p$ spaces. Compact support is important for other reasons but it starts to involve topics that are not strictly measure-theoretic in nature (e.g. distribution theory).
Your other question (how "big" can the boundary of the support of a continuous function be?) should probably be asked separately.
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