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$\Omega \subset \mathbb{R}^{n}$ is an open subset.
> [!theorem] Theorem. ([[fundamental lemma of the calculus of variations]])
> [[locally integrable function|If]] $g \in L^{1}_{\text{Loc}}(\Omega)$ is such [[continuously differentiable|that]] $\int _{\Omega} g \varphi \, =0$ [[compact|for]] [[support|all]] $\varphi \in C_{c}^{\infty}(\Omega)$, then $g = 0$ [[almost-everywhere|a.e.]] in $\Omega$.
^theorem
> [!proof]- Proof. ([[fundamental lemma of the calculus of variations]])
>
> We only consider the case that $\Omega$ is [[bounded set|bounded]] and $g \in L^{1}(\Omega)$. The general case (exercise) comes from truncation.[^2]
>
> It suffices to show $\|g\|_{1}=0$.
>
> If $g \in C^{\infty}_{c}(\Omega)$, then considering $\varphi=g$ witnesses the result. In general $g$ is not a test function; but it can be made arbitrarily close to one.
>
> [[Lp density of compactly supported functions|Recalling]] [[dense|that]] $\overline{C_{c}^{\infty}(\Omega)}=L^{1}(\Omega)$, obtain a test function $h$ "arbitrarily close to $g
quot;. More precisely, for all $\varepsilon>0$, there exists $h=h_{\varepsilon} \in C_{c}^{\infty}(\Omega)$ such that $\|g-h\|_{1}<\varepsilon$. Hence $\|h\|_{1}\geq \|g\|_{1}-\varepsilon. \ \ (*)$Want $\|h\|_{1}$ very small.
>
> For $\delta>0$, let $h_{\delta}:= \frac{h}{\sqrt{ \delta^{2} + h ^{2} }}$, so that $h_{\delta} \in C^{\infty}_{c}(\Omega)$ and $|h_{\delta}| \leq 1$. ($\delta$ is there to prevent division by zero.) [[Hölder's inequality|Now]], $|\int _{\Omega} \, gh_{\delta}- \int _{\Omega} \, h h_{\delta}|=|\int _{\Omega} (g-h) h _{\delta} \, | \leq \underbrace{ \|g-h\|_{1} }_{ \leq \varepsilon } \underbrace{ \|h_{\delta}\|_{\infty} }_{ \leq 1 } \leq \varepsilon.$By hypothesis, $\int _{\Omega} g h_{\delta} =0$. Also, $|h h_{\delta}| \uparrow |h |$ as $\delta \to 0$. Invoking [[monotone convergence theorem for nonnegative measurable functions|MCT]], we see $\varepsilon \geq \lim_{\delta \to 0} \int _{\Omega} | h h_{\delta} | \, = \int _{\Omega} |h| \, d\mu =\|h\|_{1}.$
> Combining with $(*)$, gives $\|g\|_{1} \leq 2 \varepsilon$ for all $\varepsilon>0$, and the result follows.
>
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####
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#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```