Uniform Convergence of Integrable Functions and Counterexamples [closed]

Let $ \{ f_n \}_{n=1}^{\infty} $ be a sequence of real-valued integrable functions on $ X $ such that$\sup_{x \in X} |f_n(x) - f(x)| \to 0 \quad \text{as } n \to \infty$(that is, $ f_n $ converges to $ f $ uniformly on $X$ ). Then $ f$ is integrable and$\int_X f_n \, d\mu \to \int_X f \, d\mu \quad \text{as } n \to \infty.$Give an example that shows that the claim can fail.

Maybe this statement will fail when $\mu(X)=\infty$, but I can not construct one.