1 The Riemann function $f:[-1,1]\longrightarrow \mathbb{R}$ defined as
$$ f(x)= \begin{cases} 0, &\textrm{if }x \notin \mathbb{Q}\cap[-1,1]\textrm{ or }x =0\\ \frac{1}{q} , &\textrm{if }x\in \mathbb{Q} \cap[-1,1]\textrm{ and }x =\frac{p}{q}\neq 0 \end{cases} $$
Where $x =\frac{p}{q}$ is an irreducible fraction and q is non zero ($q\neq o)$
Does the Lebesgue Criterion for Riemann Integrability hold for this function?
[2] If $f_n \longrightarrow f$ almost everywhere and $g_n \longrightarrow g$ almost everywhere where $f,g$ are both finite valued, then show that
$f_n+g_n \longrightarrow f+g$ almost everywhere And
$f_ng_n \longrightarrow fg$ almost everywhere
MY ATTEMPT
1 YES!
$f$ is discontinuous at any rational point $x =\frac{p}{q}$ $(q \neq0)$ and continuous at any irrational point in [-1,1] and f is bounded.
Since $A= \mathbb{Q} \cap[-1,1]$ -{0}the set of discontinuous points of f. Hence $m(A)=0$
Therefore by Lebesgue Criterion for Riemann integrability $f$ is Riemann integrable on [-1,1] Hence $f$ is Lebesgue Integrable and
$\int^{1}_{-1} f(x) dx = \int_{[-1,1]} fdm=0$ because $f=0$ almost everywhere
IS this correct?
[2] I'm not sure how to approach this question. I've tried using THIS
but I am unable to see the connection if there is any at all. Never did a question like this so i'm unsure as to how to begin.