[proofplan]
Each property follows from a direct computation using the Calderón-Zygmund average bounds. For (i): off $\Omega$ the value is $\le \lambda$ by the off-set bound of the [Calderón-Zygmund Decomposition](/theorems/3154); on $Q_j$ the value is the average $f_{Q_j}$, bounded by $2^n\lambda$ by the upper average bound. For (ii): the support is built into the definition $b_j = (f - f_{Q_j})\mathbb{1}_{Q_j}$, and the mean-zero condition is the defining property of subtracting the average. For (iii): triangle inequality and the average upper bound. For (iv): the goodness norm decomposes into off-$\Omega$ and on-$\Omega$ pieces; the badness norm is bounded by triangle and (iii) summed over the disjoint cubes.
[/proofplan]
[step:Bound $g$ pointwise by $2^n \lambda$ to obtain $\|g\|_{L^\infty} \le 2^n \lambda$]
Off the bad set $\Omega$, $g(x) = f(x)$. By the off-set bound of the [Calderón-Zygmund Decomposition](/theorems/3154), $|f(x)| \le \lambda$ for $\mathcal{L}^n$-a.e. $x \notin \Omega$, so
\begin{align*}
|g(x)| = |f(x)| \le \lambda \le 2^n \lambda \quad \text{for $\mathcal{L}^n$-a.e. } x \notin \Omega.
\end{align*}
On each cube $Q_j \subseteq \Omega$, $g(x) = f_{Q_j}$ is constant. By Jensen's inequality (or directly by the triangle inequality applied to the integral),
\begin{align*}
|f_{Q_j}| = \left|\frac{1}{|Q_j|}\int_{Q_j} f\, d\mathcal{L}^n(y)\right| \le \frac{1}{|Q_j|}\int_{Q_j}|f|\, d\mathcal{L}^n(y) \le 2^n \lambda,
\end{align*}
where the last inequality is the average upper bound from (ii) of the [Calderón-Zygmund Decomposition](/theorems/3154). Hence $|g(x)| \le 2^n \lambda$ for every $x \in \Omega$ as well, and combining:
\begin{align*}
\|g\|_{L^\infty(\mathbb{R}^n)} = \operatorname{ess\,sup}_{x \in \mathbb{R}^n} |g(x)| \le 2^n \lambda,
\end{align*}
which is (i).
[/step]
[step:Verify the support condition and the mean-zero cancellation for each $b_j$]
By definition $b_j = (f - f_{Q_j})\mathbb{1}_{Q_j}$, so $b_j(x) = 0$ for every $x \notin Q_j$. Hence $\{b_j \ne 0\} \subseteq Q_j$, and taking closures (since support is the closure of the non-zero set),
\begin{align*}
\operatorname{supp} b_j = \overline{\{b_j \ne 0\}} \subseteq \overline{Q_j}.
\end{align*}
For the mean-zero property,
\begin{align*}
\int_{Q_j} b_j\, d\mathcal{L}^n(y) = \int_{Q_j} (f - f_{Q_j})\mathbb{1}_{Q_j}\, d\mathcal{L}^n(y) = \int_{Q_j} f\, d\mathcal{L}^n(y) - f_{Q_j}|Q_j|.
\end{align*}
By the definition $f_{Q_j} = \frac{1}{|Q_j|}\int_{Q_j} f\, d\mathcal{L}^n(y)$, the right-hand side is
\begin{align*}
\int_{Q_j} f\, d\mathcal{L}^n(y) - \left(\frac{1}{|Q_j|}\int_{Q_j} f\, d\mathcal{L}^n(y)\right)|Q_j| = 0,
\end{align*}
proving (ii).
[/step]
[step:Estimate $\|b_j\|_{L^1}$ via triangle inequality and the cube-average upper bound]
By the triangle inequality and $|\mathbb{1}_{Q_j}| \le 1$,
\begin{align*}
\|b_j\|_{L^1} = \int_{\mathbb{R}^n} |b_j|\, d\mathcal{L}^n(y) = \int_{Q_j}|f - f_{Q_j}|\, d\mathcal{L}^n(y) \le \int_{Q_j}|f|\, d\mathcal{L}^n(y) + |f_{Q_j}|\,|Q_j|.
\end{align*}
By the same Jensen step as before, $|f_{Q_j}| \le \frac{1}{|Q_j|}\int_{Q_j}|f|\, d\mathcal{L}^n(y)$, so $|f_{Q_j}||Q_j| \le \int_{Q_j}|f|\, d\mathcal{L}^n(y)$. Substituting,
\begin{align*}
\|b_j\|_{L^1} \le \int_{Q_j}|f|\, d\mathcal{L}^n(y) + \int_{Q_j}|f|\, d\mathcal{L}^n(y) = 2\int_{Q_j}|f|\, d\mathcal{L}^n(y).
\end{align*}
Combining with the upper average bound $\frac{1}{|Q_j|}\int_{Q_j}|f|\, d\mathcal{L}^n(y) \le 2^n\lambda$ from (ii) of the [Calderón-Zygmund Decomposition](/theorems/3154),
\begin{align*}
\|b_j\|_{L^1} \le 2 \int_{Q_j}|f|\, d\mathcal{L}^n(y) \le 2 \cdot 2^n \lambda |Q_j| = 2^{n+1}\lambda |Q_j|,
\end{align*}
which is (iii).
[/step]
[step:Bound $\|g\|_{L^1}$ by decomposing the integration domain into $\mathbb{R}^n \setminus \Omega$ and the cubes]
The function $g$ is integrable as the sum of two pieces. By disjointness of $\mathbb{R}^n \setminus \Omega$ and $\Omega = \sqcup_j Q_j$ (the union being disjoint by the [Calderón-Zygmund Decomposition](/theorems/3154)),
\begin{align*}
\|g\|_{L^1} = \int_{\mathbb{R}^n}|g|\, d\mathcal{L}^n(y) = \int_{\mathbb{R}^n \setminus \Omega}|f|\, d\mathcal{L}^n(y) + \sum_{j \in J} \int_{Q_j} |f_{Q_j}|\, d\mathcal{L}^n(y).
\end{align*}
The first term is the $L^1$ norm of $f$ restricted to $\mathbb{R}^n \setminus \Omega$. For the second, $|f_{Q_j}|$ is constant on $Q_j$, so $\int_{Q_j}|f_{Q_j}|\, d\mathcal{L}^n(y) = |f_{Q_j}||Q_j|$, and the Jensen bound gives $|f_{Q_j}||Q_j| \le \int_{Q_j}|f|\, d\mathcal{L}^n(y)$. Summing,
\begin{align*}
\|g\|_{L^1} \le \int_{\mathbb{R}^n \setminus \Omega}|f|\, d\mathcal{L}^n(y) + \sum_{j \in J} \int_{Q_j}|f|\, d\mathcal{L}^n(y) = \int_{\mathbb{R}^n}|f|\, d\mathcal{L}^n(y) = \|f\|_{L^1},
\end{align*}
where countable additivity of the integral on the disjoint partition $\mathbb{R}^n = (\mathbb{R}^n \setminus \Omega) \sqcup \bigsqcup_j Q_j$ collapses the sum to the global integral. This proves the first half of (iv).
[/step]
[step:Bound $\|b\|_{L^1}$ by summing the per-cube estimates]
By disjointness of the supports and triangle inequality applied to the (countable) sum $b = \sum_j b_j$,
\begin{align*}
\|b\|_{L^1} = \int_{\mathbb{R}^n}\left|\sum_{j \in J} b_j\right|\, d\mathcal{L}^n(y) = \sum_{j \in J} \int_{Q_j} |b_j|\, d\mathcal{L}^n(y) = \sum_{j \in J} \|b_j\|_{L^1},
\end{align*}
where the second equality uses that the supports $Q_j$ are pairwise disjoint, so $|\sum_j b_j| = \sum_j |b_j|$ pointwise. Applying the per-cube bound from the third step,
\begin{align*}
\|b\|_{L^1} = \sum_{j \in J}\|b_j\|_{L^1} \le \sum_{j \in J} 2 \int_{Q_j}|f|\, d\mathcal{L}^n(y) = 2\int_{\bigcup_j Q_j}|f|\, d\mathcal{L}^n(y) \le 2\int_{\mathbb{R}^n}|f|\, d\mathcal{L}^n(y) = 2\|f\|_{L^1},
\end{align*}
which is the second half of (iv). Combining with (i)-(iii) above completes the proof.
[/step]
