# Lower semi-continuity of integration

I've found many papers characterizing the weak lower semi-continuity of $$\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,$$ on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$. Are there any results with regards to the same statement with $\Omega$ being all of $\mathbb{R}^d$ itself?

Here $f$ is a non-negative, Borel measurable function, which is convex in its second variable for a.e. one of its first variagbles.

• You should clarify what statement you have in mind. For example $\int_{\mathbb{R}^n}|u|^2$ is (sequentially) weakly lower semicontinuous since the norm is (sequentially) weakly lower semicontinuous. – Piotr Hajlasz Jun 20 '18 at 22:42
• Aren't sequential weak lower-semi-continuity and weak lower-semi-continuity the same in a Hilbert space? – AIM_BLB Jun 20 '18 at 22:50
• What are you assuming about $f$? If it's continuous in the second variable and nonnegative then this is just Fatou's lemma. And if it's not continuous then I don't see why you would expect any semicontinuity at all. – Nate Eldredge Jun 20 '18 at 23:02
• Even under those assumptions, whouldn't that only give strong lower semi continuity which is less than weak lower semi continuity? – AIM_BLB Jun 20 '18 at 23:05
• Oh yeah, I missed "weak". But Borel is certainly not enough; take $f(x,u) = 1_{[0,1]}(u)$ and $u_n = 1+1/n$. – Nate Eldredge Jun 20 '18 at 23:07