= Solution
For $1\leq p<\infty$, the real <Lp space> is the <vector space> of real <measurable functions> with $\int_\Omega|f|^p\,d\mu<\infty$, identifying functions equal <almost everywhere>. Its <Lp norm> is $\|f\|_p=(\int|f|^p\,d\mu)^{1/p}$. For $p=\infty$, take the essentially bounded real <measurable functions>, with the same identification and <norm> $\|f\|_\infty=\operatorname{ess\,sup}|f|$. The identification makes each <norm> definite; absolute homogeneity follows from the <Lebesgue integral>, and the <triangle inequality> follows from <Minkowski inequality> for finite $p$, or directly from the <essential supremum> for $p=\infty$.
Here is a <completeness> proof valid on any <measure space>. For $p<\infty$, a <Cauchy sequence> $(f_n)$ has a <subsequence> $(f_{n_k})$ with $\|f_{n_{k+1}}-f_{n_k}\|_p\leq2^{-k}$. Choose <measurable function> representatives and put $h_k=f_{n_{k+1}}-f_{n_k}$. By <Minkowski inequality> and the <monotone convergence theorem>,
$$
\left\|\sum_{k=1}^N|h_k|\right\|_p\leq\sum_{k=1}^N2^{-k}\leq1,
\qquad
\int\left(\sum_{k=1}^\infty|h_k|\right)^p\,d\mu\leq1.
$$
Consequently the <series> $\sum h_k$ converges absolutely <almost everywhere>. Define $f=f_{n_1}+\sum h_k$ there, and define it to be zero on the measurable exceptional null set. Then $f\in L^p$, and <Fatou lemma> applied to each tail gives $\|f-f_{n_k}\|_p\leq\sum_{j=k}^\infty2^{-j}\to0$. The original <Cauchy sequence> also converges in <Lp norm>, by the <triangle inequality>. For $p=\infty$, choose the same <subsequence> using the <essential supremum> <norm>. Outside one measurable null set, all the bounds $|h_k|\leq2^{-k}$ hold and $f_{n_1}$ is bounded. The <series> then converges uniformly there, with an essentially bounded <measurable function> limit and the same tail estimate in <essential supremum> <norm>. Thus \b[all these spaces are <Banach spaces>].
The <duality of Lp spaces> says that, for $1<p<\infty$ and $q=p/(p-1)$, the map
$$
I_p:L^q\longrightarrow(L^p)^*,\qquad (I_pg)(f)=\int fg\,d\mu
$$
is an <isometric isomorphism of normed spaces>. This form of <Lp duality on an arbitrary measure space> requires no finiteness hypothesis on $\mu$. At $p=1$, a standard version assumes a <sigma-finite measure> and identifies $(L^1)^*$ with $L^\infty$ through the same <dual pairing>. That endpoint assertion must not be made without a suitable measure-space hypothesis.
We first prove the required <duality of Lp spaces> for a <finite measure>. Let $F\in(L^p)^*$ and set $\nu(E)=F(\mathbf1_E)$. For disjoint measurable $E_j$, the <indicator functions> of their partial unions converge in <Lp norm> to that of their union, so $\nu$ is countably additive. It has finite <variation measure>: for every finite measurable partition $(E_j)$, choosing real signs gives
$$
\sum_j|\nu(E_j)|=F\left(\sum_j\operatorname{sgn}(\nu(E_j))\mathbf1_{E_j}\right)
\leq\|F\|\mu(\Omega)^{1/p}.
$$
Also $\mu(E)=0$ implies $\nu(E)=0$. The <Radon-Nikodym theorem> supplies a <Radon-Nikodym derivative> $g\in L^1$ with $\nu(E)=\int_Eg\,d\mu$. Linearity gives $F(f)=\int fg\,d\mu$ for <simple functions>. Uniform approximation by <simple functions> extends this identity to bounded <measurable functions>: both their <Lp norm> errors and the errors in integration against $g$ tend to zero.
To establish the correct <integrability>, test with the bounded <measurable function> $f_N=\operatorname{sgn}(g)|g|^{q-1}\mathbf1_{\{|g|\leq N\}}$. Since $(q-1)p=q$, writing $A_N=\int_{\{|g|\leq N\}}|g|^q\,d\mu$ gives
$$
A_N=F(f_N)\leq\|F\|A_N^{1/p},\qquad A_N^{1/q}\leq\|F\|.
$$
The second inequality is also valid when $A_N=0$. The <monotone convergence theorem> gives $g\in L^q$ and $\|g\|_q\leq\|F\|$. Density of <simple functions> in the <Lp space>, together with <Hölder's inequality>, now gives $F=I_pg$ on all of $L^p$. Conversely <Hölder's inequality> gives $\|I_pg\|\leq\|g\|_q$. If $g\ne0$, testing against
$$
f=\frac{\operatorname{sgn}(g)|g|^{q-1}}{\|g\|_q^{q-1}}
$$
gives $\|f\|_p=1$ and $(I_pg)(f)=\|g\|_q$, so \b[$\boxed{\|I_pg\|=\|g\|_q}$]. This also proves uniqueness of the representing <Radon-Nikodym derivative>.
For completeness, the passage to an arbitrary <measure space> can be made without losing a hypothesis in the <reflexive Banach space> argument below. On a <sigma-finite measure> space, exhaust by nested finite-measure sets $E_n$. The representing <Radon-Nikodym derivatives> on $E_n$ agree on overlaps by uniqueness. Their glued density has <Lp norm> $\|g\|_q\leq\|F\|$ by the <monotone convergence theorem>, and represents $F$ because $f\mathbf1_{E_n}\to f$ in <Lp norm>. Every $f\in L^p$ on an arbitrary <measure space> is supported on a sigma-finite measurable set: the sets $\{|f|>1/n\}$ have finite measure and their union is $\{f\ne0\}$.
Use <support localization of an Lp functional> as follows. For a sigma-finite measurable $A$, let $m_A$ be the <operator norm> of $F$ restricted to functions supported in $A$. The support observation gives $\sup_A m_A=\|F\|$. Choose $A_n$ approaching this supremum and set $A=\bigcup_nA_n$; then $m_A=\|F\|$. If a sigma-finite $B\subseteq\Omega\setminus A$ had $m_B=\beta>0$, functions supported on the disjoint sets $A,B$ have the direct-sum <Lp norm>. Optimizing their two scalar coefficients by <Hölder's inequality>, and using functions approaching the two restriction <operator norms>, would give
$$
\|F\|\geq(m_A^q+\beta^q)^{1/q}>m_A,
$$
a contradiction. Thus $F$ vanishes on functions supported outside $A$. The density on $A$, extended by zero, represents $F$ globally. The case $F=0$ simply uses $g=0$. This proves the stated <Lp duality on an arbitrary measure space>.
Finally let $X=L^p$ and let $\Phi\in X^{**}$, where stars denote <continuous dual spaces>. Compose with $I_p$ to obtain the <bounded linear functional> $g\mapsto\Phi(I_pg)$ on $L^q$. Applying <duality of Lp spaces> with the exponents reversed gives $f\in L^p$ with $\Phi(I_pg)=\int fg\,d\mu$. For the <canonical embedding into the bidual> $J_Xf$, its value on $I_pg$ is also $\int fg\,d\mu$. Since $I_p$ is onto, $\Phi=J_Xf$. Its <norm> is $\|f\|_p$ by the same <dual pairing> <norm> identity. Therefore \b[$\boxed{J_X(L^p)=(L^p)^{**}}$], which proves that $L^p$ is a <reflexive Banach space> through its actual <canonical embedding into the bidual>.
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