= Solution
For a locally compact Hausdorff Abelian group $G$, <Bochner's theorem> states that a continuous <positive-definite function> $p$ has a unique representation
$$
\boxed{p(x)=\int_{\widehat G}\chi(x)\,d\nu(\chi),\qquad
\nu\ge0,\quad\nu(\widehat G)=p(e)},
$$
where $\nu$ is a finite <Radon measure> on the <Pontryagin dual group>. Conversely every such measure gives a continuous <positive-definite function>. Here positive definiteness means $\sum_{i,j}c_i\overline{c_j}p(x_ix_j^{-1})\ge0$ for all finite choices. The Abelian and continuity hypotheses are part of the theorem; the representation is not claimed for arbitrary non-Abelian groups or discontinuous functions.
First prove existence without presupposing the representation. The elementary consequences of positive definiteness give $p(x^{-1})=\overline{p(x)}$, $p(e)\ge0$ and $|p(x)|\le p(e)$. If $p(e)=0$, then $p=0$ and take $\nu=0$. Otherwise form finite formal linear combinations of symbols $[x]$, with sesquilinear form
$$
\langle[x],[y]\rangle=p(xy^{-1}),
$$
linear in its first variable. Positive definiteness makes this form <positive semidefinite>. The <Cauchy-Schwarz inequality> for such a form shows that null vectors are orthogonal to everything, so quotienting by them gives an <inner product space>. Complete it to a <Hilbert space> $H$. Since $G$ is Abelian, translation $U_a[x]=[ax]$ preserves the form and extends to a <unitary operator>. Its inverse is $U_{a^{-1}}$, so $(U_a)$ is a <unitary representation>. With $v=[e]$,
$$
p(x)=\langle U_xv,v\rangle,\qquad\|v\|^2=p(e).
$$
The translated copies of $v$ span a dense subspace. Moreover
$$
\|U_a[y]-[y]\|^2=2[p(e)-\operatorname{Re}p(a)]\longrightarrow0\quad(a\to e).
$$
Finite sums and density, together with $\|U_a\|=1$, give <strong continuity> on all of $H$. This constructs the <cyclic unitary representation of a positive-definite function> directly.
For $f\in L^1(G)$ define the <integrated unitary representation> $\pi(f)=\int f(x)U_x\,dx$, with the integral applied to each vector. Its <norm> is at most $\|f\|_1$. Fubini gives $\pi(f*g)=\pi(f)\pi(g)$, and inversion in the Abelian group gives $\pi(f^*)=\pi(f)^*$. Thus
$$
\mathcal A=\overline{\pi(L^1(G))}^{\|\cdot\|}
$$
is a commutative <C-star algebra>. It is nondegenerate: choose nonnegative compactly supported functions $u_V$ of Haar integral one with supports shrinking to the <identity element>. <Strong continuity> implies
$$
\|\pi(u_V)w-w\|\le\sup_{x\in V}\|U_xw-w\|\longrightarrow0
$$
for every $w\in H$. These form an <approximate identity> in the representation. <Neighbourhoods> may form a net, since no metrizability is being assumed here.
The <Commutative Gelfand--Naimark theorem> identifies $\mathcal A$ with $C_0(X)$, where $X$ is its <character space>. The positive vector functional $\omega(B)=\langle Bv,v\rangle$ becomes a bounded positive functional on $C_0(X)$, so the <Riesz-Markov-Kakutani representation theorem> supplies a finite positive <Radon measure> $\rho$ with
$$
\omega(B)=\int_X\theta(B)\,d\rho(\theta).
$$
These general representation theorems are applied to the explicitly constructed algebra; they are not an assumption of Bochner's conclusion.
For each $\theta\in X$, the functional $f\mapsto\theta(\pi(f))$ is a nonzero <algebra character> of $L^1(G)$, since $\pi(L^1(G))$ is dense in $\mathcal A$. By Question 3 it has the form $\int f(x)\chi_\theta(x)\,dx$. The resulting map $\iota:X\to\widehat G$ is continuous: pointwise convergence of $\theta$ on $\mathcal A$ gives convergence on all $\pi(f)$, and the topological character identification in Question 3 gives compact-open convergence of $\chi_\theta$. Push $\rho$ forward under $\iota$ to obtain $\nu$. This <finite measure> is Radon: <compact subsets> approximating the measure of any inverse-image <Borel set> have compact images, giving inner regularity; finiteness and complements then give outer regularity.
To recover $p(x)$, note that $U_x\pi(u_V)=\pi(T_xu_V)\in\mathcal A$. Hence
$$
\langle U_x\pi(u_V)v,v\rangle
=\int_X\chi_\theta(x)\Phi_{\chi_\theta}(u_V)\,d\rho(\theta).
$$
The left side tends to $p(x)$. On the right, $|\Phi_\chi(u_V)|\le1$ and $\Phi_\chi(u_V)\to1$ uniformly on <compact subsets> of $\widehat G$. For completeness, the evaluation $(\chi,x)\mapsto\chi(x)$ is jointly continuous: near $x_0$ restrict $x$ to a compact <neighbourhood>, control the character uniformly there, and use continuity of the fixed character. A finite cover of a <compact set> of characters then makes those characters equicontinuous at $e$, proving the stated <uniform convergence>. Continuity of $\iota$ gives <uniform convergence> on <compact subsets> of $X$, and the finite <Radon measure> $\rho$ makes the remaining tails arbitrarily small. Thus passage to the limit is valid even for a net, and
$$
p(x)=\int_X\chi_\theta(x)\,d\rho(\theta)=\int_{\widehat G}\chi(x)\,d\nu(\chi).
$$
Taking $x=e$ also gives the required total mass $p(e)$.
Conversely, a finite positive <Radon measure> gives
$$
\sum_{i,j}c_i\overline{c_j}p(x_ix_j^{-1})
=\int_{\widehat G}\left|\sum_i c_i\chi(x_i)\right|^2d\nu(\chi)\ge0.
$$
Its transform is continuous. Indeed, choose a <compact set> of characters with arbitrarily small measure outside it; joint continuity of evaluation gives uniform continuity in $x$ near the specified point on that <compact set>, while the outside contribution is bounded by twice its measure. This proves continuity without using an unjustified dominated-convergence assertion for arbitrary nets.
Finally suppose two finite <Radon measures> have the same transform. Multiply by $f\in L^1(G)$ and integrate over $G$. Fubini shows that the measures agree on every function $\chi\mapsto\Phi_\chi(f)$. Question 3 showed that these functions are uniformly dense in $C_0(\widehat G)$. Boundedness of both measure functionals extends equality to all of $C_0(\widehat G)$; uniqueness in the Riesz representation theorem gives equality of the measures. This proves existence, uniqueness and the converse, completing Bochner's theorem. In particular \b[$p(e)=1$ corresponds exactly to a <probability measure>] on the dual group.
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