Let $\{f_n \}_{n \geq 1}$ and $f$ be integrable functions on $[0,1]$ such that $$\int_{0}^{1} f_n (x)\ dx \to \int_{0}^{1} f(x)\ dx\ \text {as}\ n \to \infty.$$
Does it imply that $f_n (x) \to f(x),$ as $n \to \infty$ for almost every $x \in [0,1]\ $?
No, this doesn't hold. Integrals are aggregates, and you therefore lose information about the functions by integrating; saying that two curves have similar area under them hardly means the curves themselves are similar.
For instance: take $f(x)=1$, and take $f_n(x)=2x$ for all $n$.
Then $\int_0^1f_n(x)\,dx=\int_0^1 f(x)\,dx=1$ for all $n$, but clearly $f_n$ does not converge to $f$ pointwise almost everywhere.