Prove or disprove that $\forall d\ge_\mathbb Z0,\forall f\colon\mathbb R^d\to[0,+\infty],$ $$\lim_{n\to+\infty}\overline{\int_{\mathbb R^d}}\min(f(x),n)\,\mathrm{d}x=\overline{\int_{\mathbb R^d}}f(x)\,\mathrm{d}x.$$
I tried, unsuccessfully, to prove it similar to how I proved the lower signed Lebesgue integral version.
This is from Tao's "An Introduction to Measure Theory" page 54.
Edit 1. The upper signed Lebesgue integral is defined as $$\overline{\int_{\mathbb R^d}}f(x)\,\mathrm{d}x:=\inf _{h \geq f;\,h\text{ simple}} \operatorname{Simp} \int_{\mathbb{R}^{d}} h(x)\,\mathrm{d}x.$$