Let $\Omega =(0,1)$, and $f,f_n\in \mathcal{L}_p(\lambda)$.
Prove that if $\sup_n{\| f_n \|}<\infty$ and $$\int_{(0,t]}f_n \, \,\mathrm{d}\lambda\rightarrow \int_{(0,t]}f \, \,\mathrm{d}\lambda,$$
for all $t\in(0,1)$, then $$\int_{E}f_n \, \,\mathrm{d}\lambda\rightarrow \int_{E}f \, \,\mathrm{d}\lambda,$$ for all $E$ in the Borel $\sigma$-algebra for $(0,1).$
And that this in turn implies $f_n \rightarrow f$ in the weak topology; i.e. for all $g\in \mathcal{L}_q$,
$$\int_\Omega g f_n \, \,\mathrm{d}\lambda\rightarrow \int_{\Omega}gf \, \,\mathrm{d}\lambda.$$
For the first part I was thinking of using the regularity of $\lambda$ to approximate $E$ by intervals $(0,t]$ but I don't know how to proceed, or even if I'm on the right track (I haven't dealt with the Lebesgue measure in a long time).
Edit: I got the second part already, approximating $g$ by a simple function and using the hypothesis on the converging of integrals for measurable sets. It is only the first part that I don't know how to attack.
Any pointers would be greatly appreciated.
Use that $\chi_{(a,b]} = \chi_{(0,b]} - \chi_{(0,a]}$ for $a<b$. This will show that your first assumption yields
$$ \int f_n \cdot \chi_{(a,b]} \to \int f \cdot \chi_{(a,b]} $$ for all $a<b$. By linearity, we get the same convergence for arbitrary Riemann step functions, i.e. for step functions with respect to intervals.
Now, one possibility is to use that $C_c ((0,1))$ is dense in $L^p ((0,1))$ and that you can approximate each function $f \in C_c$ uniformly (and hence also in $L^p$) by Riemann step functions as above.
Then conclude the proof by density as you did for the second claim.