Is this functional $\psi(u)=\int_{\Omega} \int_{0}^{u(x)} H(u(x)-\mu)dsdx$ is upper semi-continuous?

Question

Let $X$ be a real Banach space, for $u \in X$ we define the following functional $$\psi(u)=\int_{\Omega} \int_{0}^{u(x)} H(u(x)-\mu)dsdx$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ containing the origin, $\mu$ is a positive real parameter and $H$ is the Heaviside function i.e. $$H(t)=\left\{\begin{array}{ll} 1 & \quad \mbox{if }\ t\geq 0, \\[0.1cm] 0 & \quad \mbox{if }\ t<0 . \end{array} \right.$$ Can the following functional be upper semi-continuous given that the Heaviside function is discontinuous? and if the answer is yes, how to verify the latter?

Answer 1

Let $\Omega=[0,1]$ and $X=L_\nu^1([0,1])$ where $\nu$ is the measure given by $\nu(A)=\int_At^2dt$, then $n\chi_{[0,\frac{1}{n}]}$ converges to $0$ in $X$ but for $\mu<1$ the functional converges to $+\infty> \psi(0)$.