Bounding 2d functions by products of 1d functions

Given sets X,Y$X,Y$ and a regular cardinal mu, let phi(X,Y,mu)$\Phi (X,Y,\mu )$ be the statement that for any function f:XxY ->mu$f : X \times Y \rightarrow \mu$, there are functions g1:X ->mu$g_1 : X \rightarrow \mu$ and g2:Y ->mu$g_2 : Y \rightarrow \mu$ such that for all (x,y)is an element of XxY$(x,y) \in X \times Y$, f(x,y)<= max{g1(x),g2(y)}$f(x,y) \le \max \lbrace g_1(x), g_2(y) \rbrace$. In ZFC$\mathsf {ZFC}$, the statement phi(omega 1,omega 1,omega)$\Phi (\omega _1, \omega _1, \omega )$ is false. However, we show the theory ZF+"theclubfilteron omega 1isnormal"+phi(omega 1,omega 1,omega)$\mathsf {ZF}+ \text{``the club filter on $\omega _1$ is normal''} + \Phi (\omega _1, \omega _1, \omega )$ (which is implied by ZF+DC$\mathsf {ZF}+ \mathsf {DC}$+ "V=L(R)$V = L(\mathbb {R})$" + "omega(1) is measurable") implies that for every alpha<omega 1$\alpha < \omega _1$ there is a kappa is an element of(alpha,omega 1)$\kappa \in (\alpha ,\omega _1)$ such that in some inner model, kappa is measurable with Mitchell order >=alpha$\ge \alpha$.