Lebesgue integral that satisfies $\int_X u=2017$ and $\lim_{n\to\infty}\int_X u_n=2018$

Question

I have to find a sequence of functions that satisfies $\int_X xd\mu=2017$ and $\lim_{n\to\infty}\int_X x_nd\mu=2018$ given $(X,A,\mu)$ is a measurable space $\mathcal{M}^+_{\bar{\mathbb{R}}}$ and $x_n\to x$ for $n\to\infty$ (pointwise convergence).

I know from Fatou's lemma that such function exists as it says: $$\int_X xd\mu\leq \lim_{n\to\infty}\int_X x_nd\mu,$$ but I cannot really find such function when $x$ has to be convergent.

Answer 1

You can have $x_n \to x$ uniformly if you like. Define $x_n(t) = \dfrac 1n \chi_{[n,2n)}(t)$. Then $x_n \to 0$ uniformly but $\displaystyle \int_{\mathbb R} x_n(t) \, dt = 1$ for all $n$.