= Solution
For $t\in\mathbb R$, define the set functional
$$
F_t(E)=\alpha\operatorname{Per}(E;\Omega)+\int_E(t-f)\,dx.
$$
The <ROF level-set formulation> states that the unique <minimizer> of $J(u)=\|u-f\|_2^2/2+\alpha\operatorname{TV}(u)$ has <superlevel sets> minimizing $F_t$ for almost every $t$. Conversely, an admissible $u$ with minimizing <superlevel sets> is the ROF <minimizer>. Thus the concise characterization is
$$
\boxed{\hat u\text{ minimizes ROF}\iff\{\hat u>t\}\in\arg\min_E F_t(E)\text{ for a.e. }t.}
$$
The <signed layer-cake identity for quadratic fidelity> and the <coarea formula for BV functions> give
$$
J(u)-\frac12\|f\|_2^2=\int_{\mathbb R}\bigl[F_t(\{u>t\})-F_t(E_t^0)\bigr]dt,\qquad E_t^0=\begin{cases}\Omega,&t<0,\\\varnothing,&t\ge0.\end{cases}
$$
Indeed $u^2/2-fu=\int_{\mathbb R}(t-f)(\chi_{\{u>t\}}-\chi_{\{0>t\}})dt$ pointwise. Its absolute integral is bounded by $u^2/2+|fu|$, so <Fubini's theorem> applies for $u,f\in L^2$. The negative-level baseline is essential; integrating the unadjusted $F_t$ would generally diverge.
Each $F_t$ attains its minimum. A <minimizing sequence> of <indicator functions> has bounded $L^1$ <norm>, and comparison with the empty set bounds its perimeter by an $L^1$ forcing bound. <Bounded-variation compactness> supplies a limiting <indicator function>. The forcing integral converges by <dominated convergence>, while <relative perimeter> is <sequentially lower semicontinuous>.
For $s<t$, the supplied comparison lemma applied to $(f-t)/\alpha<(f-s)/\alpha$ makes every selected <minimizer> $A_t$ contained in $A_s$ up to a <null set>. The <comparison of perimeter minimizers with ordered forcing> also follows directly: compare $A_t$ with $A_t\cap A_s$, compare $A_s$ with $A_t\cup A_s$, add and use <submodularity of relative perimeter> to obtain $(t-s)|A_t\setminus A_s|\le0$.
Select <minimizers> at rational levels, remove their countably many exceptional <null sets>, and reconstruct $v(x)=\sup\{q\in\mathbb Q:x\in A_q\}$. The strict <superlevel set> is $\{v>t\}=\bigcup_{q>t}A_q$. This union also minimizes $F_t$: take $q\downarrow t$, use monotone $L^1$ convergence of its <indicator functions> and <sequential lower semicontinuity>, and note that $m(t)=\min_EF_t(E)$ satisfies $|m(t)-m(s)|\le|\Omega||t-s|$.
To justify finite energy before assuming it, clip this reconstruction to $v_M\in[-M,M]$. Comparison with the empty set gives $\alpha\operatorname{Per}(E_t;\Omega)\le\int_\Omega|t-f|$ uniformly on a bounded interval of levels. Since $v_M=-M+\int_{-M}^M\chi_{E_t}\,dt$, pairing with compactly supported test-field divergences bounds its <total variation seminorm> by $\int_{-M}^M\operatorname{Per}(E_t;\Omega)dt<\infty$. Thus $v_M$ belongs to the <BV space> before applying the <coarea formula for BV functions>. The layer-cake argument on the finite interval $[-M,M]$ shows $J(v_M)\le J(T_Mw)$ for every finite-energy competitor $w$, where $T_M$ is clipping. In particular $w=0$ bounds the $L^2$ <norms> and variations uniformly. <Fatou's lemma> excludes infinite values of $v$ on a positive-measure set. <Bounded-variation compactness> and <weak convergence> in $L^2$ identify an admissible limit $v$. For any fixed competitor, $T_Mw\to w$ in $L^2$ and its variation tends to that of $w$, by <bounded-variation contraction under clipping> and <sequential lower semicontinuity>. Thus $J(v)\le J(w)$.
The <quadratic fidelity> is <strictly convex>, so every ROF <minimizer> equals $v$ <almost everywhere> and has the selected minimizing <superlevel sets>. Conversely, for any $u$ whose levels minimize $F_t$, integrate the levelwise inequality against any competitor's levels in the displayed identity to obtain $J(u)\le J(w)$. This proves both directions for signed as well as nonnegative data.
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