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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.

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    $\begingroup$ Wait a minute... You used DCT but don't want to use measure theory arguments? (DCT is from measure theory (Lebesgue integrals) ) $\endgroup$
    – Divide1918
    Commented Apr 11, 2021 at 7:26
  • $\begingroup$ Yeah. We haven't been taught Measure Theory so far. But this is a problem which I came across. $\endgroup$
    – JRC
    Commented Apr 11, 2021 at 11:18
  • $\begingroup$ So how do you know DCT? $\endgroup$
    – Divide1918
    Commented Apr 11, 2021 at 12:52

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