I'm reading "Elliptic and Parabolic Equations" by Wu, Yin and Wang. In Section 4.2, they consider the heat equation given $u_0 \in L^\infty$ and $f \in L^\infty$ $$u_t - \Delta u = f$$ $$u(0) = u_0$$ where $u \in H^1(0,T;L^2) \cap L^2(0,T;H^1)$ on a bounded domain. Testing the equation with $(u-k)^+$, where $k$ is larger than the $L^\infty$ norm of $u_0$, and using the fact that $I_k(t) = \int_\Omega |(u(t)-k)^+|^2$ is continuous, so it has a maximum at $t=\sigma$, they derive the following:
This I don't understand. Here $\sigma$ is NOT a variable, but a particular point in $[0,T]$ which is such that $I_k(\sigma) \geq I_k(t)$ for all $t$ as I mentioned.
They seem to have used Lebesgue's differentiation theorem to obtain the final inequality but that works only for almost all $t \in [0,T]$, and $\sigma$ may lie in the null set. Is there some other way to derive this?