I am stuck on the following problem listed below from Tao's "Introduction to Measure Theory".

Exercise 11: Let $f:\mathbb{R}^d \to [0, \infty]$ be measurable, bounded, and vanishing outside of a set of finite measure. Show that the lower and upper Lebesgue integrals of f agree. (Hint: use Exercise 4.)

Here is exercise 4 for reference.

Exercise 4: Let $f:\mathbb{R}^d \to [0, \infty]$. Show that f is a bounded unsigned measurable function iff $f$ is the uniform limit of bounded simple functions.

What I have thought of so far: Since $f$ is measurable, bounded, and finitely supported, there is a sequence of bounded simple functions $\{\phi_n\}_{n = 1}^{\infty}$ such that $\phi_n \to f$ uniformly as $n \to \infty$. It would suffice to show that $$\lim_{n \to \infty}\underline{\int_{\mathbb{R^d}}} \phi_n(x) dx \to \underline{\int_{\mathbb{R^d}}} f(x) dx$$ and $$\lim_{n \to \infty}\overline{\int_{\mathbb{R^d}}} \phi_n(x) dx \to \overline{\int_{\mathbb{R^d}}} f(x) dx$$

Would this be the right approach? Usually, I would try to use one of the convergence theorems (dominated, monotone, bounded convergence), but they have not been discussed in the book and I assumed there was an simpler way to approach this problem.


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