Construct a sequence of functions $\{f_n(x)\}_{n=1}^{\infty}\subset L([0,1])$ and measurable function $f(x)$ such that $f(x)=\lim \limits_{n\to \infty} f_n(x)$ for all $x\in (0,1)$ and $$\left|\int_{(0,1)}f_n(x)d\mu\right|\leq 1$$ for all $n$, but $f(x)\notin L([0,1])$.
By $L([0,1])$ I mean Lebesgue integrable function on $[0,1]$.
I have spent some time trying to construct an example but I failed.
Would be very thankful for help!
Let $b_n := e^{-n}$, $n\in \mathbb{N} = \{0, 1, 2, \ldots\}$, and define $$ f_n(x) := \begin{cases} (-1)^j/x, & \text{if}\ b_{j+1} < x \leq b_j,\ j\leq n,\\ 0, & \text{otherwise in}\ [0,1]. \end{cases} $$ Since $\int_{b_{j+1}}^{b_j} f_n(x)\, dx = (-1)^j$, then $|\int_0^1 f_n| = 1$. On the other hand, if $f$ is the pointwise limit of the sequence $(f_n)$, then $|f(x)| = 1/x$ for every $x\in (0,1]$.