Problem 1. Let $\mathcal{A}([0,1] \times [0,1])$ be the set of all measurable functions $f:[0,1] \times [0,1] \to \mathbb{C}$ that satisfy the conditions $$ \alpha_1(f) = \operatorname*{ess sup}_{y \in [0,1]} \int\limits_{0}^{1} \lvert f(x,y) \rvert \, dx < \infty, $$ $$ \alpha_2(f) = \operatorname*{ess sup}_{x \in [0,1]} \int\limits_{0}^{1} \lvert f(x,y) \rvert \, dy <\infty. $$ One can prove that $\mathcal{A}([0,1] \times [0,1])$ is a Banach space with the norm $$ \lVert f \rVert_{\mathcal{A}([0,1] \times [0,1])} = \max \bigl(\alpha_{1}(f), \alpha_{2}(f) \bigr). $$
Question 1. Consider the set $\Phi([0,1] \times [0,1])$ of a simple functions of the form $$ \phi(x,y) = \sum_{i=1}^{N} \lambda_i \, \chi_{E_i}(x,y), $$ where $\lambda_1, \ldots \lambda_N, \in \mathbb{C}$ is a sequence of complex numbers, $E_1, \ldots, E_N$ is a sequence of disjoint measurable subsets of a set $[0,1] \times [0,1]$, $\chi_{E_i}$ is the indicator function of the set $E_i$. Is $\Phi([0,1] \times [0,1])$ dense in $\mathcal{A}([0,1] \times [0,1])$?
Question 2. Consider the set $\Psi([0,1] \times [0,1])$ of a simple functions of the form $$ \psi(x,y) = \sum_{i=1}^{N} \lambda_i \, \chi_{X_i}(x) \, \chi_{Y_i}(y), $$ where $X_1, \ldots, X_N$ and $Y_1, \ldots, Y_N$ are sequences of disjoint measurable subsets of a set $[0,1]$. Is $\Psi([0,1] \times [0,1])$ dense in $\mathcal{A}([0,1] \times [0,1])$?
Problem 2. Let $\mathcal{A}(\mathbb{R} \times \mathbb{R})$ be the set of all measurable functions $f:\mathbb{R} \times \mathbb{R} \to \mathbb{C}$ that satisfy the conditions $$ \alpha_1(f) = \operatorname*{ess sup}_{y \in \mathbb{R}} \int\limits_{-\infty}^{\infty} \lvert f(x,y) \rvert \, dx < \infty, $$ $$ \alpha_2(f) = \operatorname*{ess sup}_{x \in \mathbb{R}} \int\limits_{-\infty}^{\infty} \lvert f(x,y) \rvert \, dy <\infty. $$
It is not hard to show that $\mathcal{A}(\mathbb{R} \times \mathbb{R})$ is a Banach space with the norm \begin{equation} \lVert f \rVert_{\mathcal{A}(\mathbb{R} \times \mathbb{R})} = \max \bigl(\alpha_{1}(f), \alpha_{2}(f) \bigr). \end{equation}
Question 3. Similar to questions 1 and 2: are $\Phi(\mathbb{R} \times \mathbb{R})$ and $\Psi(\mathbb{R} \times \mathbb{R})$ dense in $\mathcal{A}(\mathbb{R} \times \mathbb{R})$?
As I can figure it out, simple functions are not dense in the Banach space of functions $f:\mathbb{R} \times \mathbb{R} \to \mathbb{C}$ that satisfy the condition $\alpha_1(f) < \infty$ only. (See A. Benedek, R. Panzone. The spaces $L^p$ with mixed norm, Duke Mathematical Journal, Volume 28, issue 3, 1961, p. 308), but I failed to understand why. And what about the space $\mathcal{A}(\mathbb{R} \times \mathbb{R})$?
My attempt to answer question 1. Let $E \subset [0,1] \times [0,1]$, and define \begin{equation*} E^x = \{y \in [0,1]:\: (x,y) \in E \}, \end{equation*} \begin{equation*} E^y = \{x \in [0,1]:\: (x,y) \in E \}. \end{equation*} Since for any $\phi \in \Phi([0,1] \times [0,1])$ \begin{equation*} \alpha_{1} (\phi) = \operatorname*{ess sup}_{y \in [0,1]} \sum_{i=1}^{N} \lambda_i \, \mu(E_i^y), \quad \alpha_{2} (\phi) = \operatorname*{ess sup}_{x \in [0,1]} \sum_{i=1}^{N} \lambda_i \, \mu(E_i^x), \end{equation*} where $\mu$ is the measure on $[0,1]$, then $\Phi([0,1] \times [0,1]) \subset \mathcal{A}([0,1] \times [0,1])$.
Suppose $f^+ \in \mathcal{A}([0,1] \times [0,1])$, $f^+(x,y) \geqslant 0$ for any $x,y \in [0,1]$ and define $$ \phi_m(x,y) = \begin{cases} \frac{k}{2^m}, &\text{if} \ \frac{k}{2^m} \leqslant f^+(x,y) < \frac{k+1}{2^m}, \ k=0,1,\ldots, 2^{m}m - 1, \\ m, &\text{if} \ f^+(x,y) \geqslant m. \end{cases} $$ Then we have $$ \forall m \in \mathbb{N}: \quad \phi_m \in \Phi([0,1] \times [0,1]), $$$$\forall x,y \in [0,1]: \quad \phi_1(x,y) \leqslant \phi_2(x,y) \leqslant \ldots \leqslant f^+(x,y), $$ and \begin{equation*} \forall x,y \in [0,1]: \quad f^+(x,y) = \lim\limits_{m \to \infty} \phi_m(x,y). \end{equation*}
From conditions $\alpha_1(f^+) < \infty$ and $\alpha_2(f^+) < \infty$ it follows that for any fixed $y \in [0,1]$ the function $f^+(\cdot, y)$ is integrable and for any fixed $x \in [0,1]$ the function $f^+(x, \cdot)$ is integrable too. Hence, the dominated convergence theorem shows that the sequences of functions \begin{equation*} g^{1}_m(y) = \int\limits_{0}^{1} \left( f^+(x, y) - \phi_m(x,y) \right) dx, \quad y \in [0,1], \end{equation*} \begin{equation*} g^{2}_m(x) = \int\limits_{0}^{1} \left( f^+(x, y) - \phi_m(x,y) \right) dy, \quad x \in [0,1], \end{equation*} converges to $0$ pointwise. Now using Egorov's theorem we can conclude that the sequences $g^{1}_m$ and $g^{2}_m$ converges to $0$ almost uniform but not uniform.
Any help would be greatly appreciated!