LOTUS for non-continuous funtion

Let $$f_n:[a,b]\to\mathbb{R}$$ be a sequence of continuous and uniformly bounded functions. Suppose, $$f_n(x)\to f(x)$$ pointwise for all $$x\in [a,b]$$ .

Now, let us take $$X\sim \text{Unif}[a,b]$$ . Then, as the $$f_n$$'s are continuous, so by the Law of the Unconscious Statistician (LOTUS), we can say that $$E[f_n(X)]=\frac{1}{b-a}\int_a^b f_n(u)~du$$ Also, due to the uniformly bounded criterion, by DCT, it can be easily shown that $$E[f_n(X)]\longrightarrow E[f(X)]$$ .

But the problem is that $$f$$ is not necessarily a continuous function. So I'm not allowed to use LOTUS on $$f$$ and write $$E[f(X)]$$ as a Riemann Integral. Can someone please help to resolve this issue without using any measure theory arguments ? You may assume that $$f$$ is Riemann integrable. Thanks in advance.

P.S. : What I want to show is : $$\int_a^b f_n(t)~dt ~\longrightarrow~\int_a^b f(t)~dt$$ where the integrals are Riemann integrals.

• Wait a minute... You used DCT but don't want to use measure theory arguments? (DCT is from measure theory (Lebesgue integrals) ) Commented Apr 11, 2021 at 7:26
• Yeah. We haven't been taught Measure Theory so far. But this is a problem which I came across.
– JRC
Commented Apr 11, 2021 at 11:18
• So how do you know DCT? Commented Apr 11, 2021 at 12:52