Find functions fitting conditions

Question

Give an example of a sequence of functions $\{u_n\}_{n \geq 1}$ that are positively measureable on $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$ that fulfills $u_n \rightarrow u$, for $n \rightarrow \infty$ and

$$ \lim_{n \rightarrow \infty} \int_{\mathbb{R}} u_n d \mu=2023, \quad \int_{\mathbb{R}} u d \mu=23 . $$ where $\mu$ is the Lebesgue measure.

Answer 1

If you think about the sequence of functions $u_{n} = 1_{[n,n+1]}(x)$ you can get the example you want. These are indicator functions on intervals of length one. Let $u(x) = 0$ and we have that $u_n \to u$ pointwise, but $\int_{\mathbb{R}} u_n(x) dx = 1$ for all $n$.

Answer 2

Let define a function

$$u_n = \mathbb{1}_{[-23, 0]} + 2000 \cdot \mathbb{1}_{[n, n+1]}$$

Clearly, for any $x$ $$ u(x) = \mathbb{1}_{[-23, 0]}(x) = \lim_{n \rightarrow \infty} u_n(x) $$

The integrals should be calculated aside.