= Solution
The answer is
$$
\boxed{p=1,\qquad p'=\infty.}
$$
Indeed, <Lebesgue measure> on $E\subset\mathbb R^n$ is a <sigma-finite measure>, and <duality of Lp spaces> gives $(L^1(E))^*=L^\infty(E)$. The restricted <Lebesgue measure> is a <non-atomic measure> and $E$ has positive measure, so the <Dual of L infinity is larger than L1>: the canonical integral representation does not exhaust $(L^\infty(E))^*$.
For $1<p<\infty$, the <Holder conjugate exponent> also lies strictly between one and infinity. <Duality of Lp spaces> then holds in both directions, in agreement with <Reflexivity of Lp spaces>. For $p=\infty$, the first required identification already fails.
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