3 Theorem. Let $p ∈[1, ∞]$. Then there exists no $h ∈ L^p((0,2))$ such that
\[
∫_{(0,2)} h ⋅ φ ⋅ d λ^1=φ(1) ∀ φ ∈ C_c^{∞}((0,2))
\]
Proof. Assume there is an $h ∈ L^p((0,2))$ with the property above.
Let $q$ be the conjugate of $p$. Further, let $φ_ϵ$ as in the lemma. Then we have
\[
\begin{aligned}
1 & =\left|φ_ϵ(1)\right| \\
& =\left|∫ h φ_ϵ d λ^1\right| ⩽ ∫|h|\left|φ_ϵ\right| d λ^1=∫\left|h 1_{B_ϵ(1)}\right|\left|φ_ϵ\right| d λ^1 \\
& ⩽\| h 1_{B_ϵ(1)}\|_p ⋅\| φ_ϵ\|_q ⩽\| h 1_{B_ϵ(1)} \|_p \\
& =\left(∫\left|h 1_{B_ϵ(1)}\right|^p d λ^1\right)^{\frac{1}{p}} \xrightarrow{ϵ → 0+} 0
\end{aligned}
\]
(note the use of Hölder's inequality; the limit follows from the dominated convergence theorem which is applicable as
\[
0 \stackrel{\text { a.e. }}{⟵}\left|h 1_{B_ϵ(1)}\right|^p ⩽|h|^p
\]
and $|h|^p$ is integrable since $h ∈ L^p((0,2))$.) Contradiction!
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