Does $f_n$ necessarily converge to f pointwise? $\int_a^b|f_n(x)-f(x)|dx → 0$ as $n → ∞$

Question

Let $(f_n)_{n=1}^∞$ with $f_n : [a,b] → ℝ$ be a sequence of Riemann integrable functions such that

$\int_a^b|f_n(x)-f(x)|dx → 0$ as $n → ∞$

for some Riemann integrable function $f : [a,b] → ℝ$. Does $f_n$ necessarily converge to $f$ pointwise?

I posted this question previously but mixed it up with another question: Does $f_n$ necessarily converge to f pointwise?

Answer 1

The answer is No because we can always change the values of all $f_n$ on a set of measure zero and the result $\int_a^b|f_n(x)-f(x)|dx → 0$ as $n → ∞$ remains valid

However if the sequence {$\{f_n\}$} is uniformly integrable then by Vitali Convergence Theorem we have $\lim_{n \to \infty}\int_a^b|f_n(x)-f(x)|dx =\int_a^b\lim_{n \to \infty}|f_n(x)-f(x)|dx =0 $ and according to Riesz theorem there is a subsequence of {$\{f_n\}$} such that $\lim_{n \to \infty}f_n=f$ almost everywhere and if $f_n$ and $f$ are both continuous then $\lim_{n \to \infty}f_n=f$ everywhere.