Approximating function by its step approximation

Question

$f$ be any bounded measurable function on $[0,1]$. Define the step function $f_n(x)=f(\frac{i}{n})$, if $x \in [\frac{i-1}{n},\frac{i}{n}]$. Is it always true that $$\int |f(x)- f_n(x)|dx \rightarrow 0?$$ Or do we need some extra condition.

Answer 1

This is not true in general. Let $f = \chi_{\mathbb{Q}^c \cap [0,1]}$. Then $f_n(x) = 0$ everywhere on $[0,1]$, for all $n$. So $\int |f(x) - f_n(x)|\,dx = \int |f(x)|\,dx = 1$.