Say $R$ is the set of all Riemann integrable functions $f:[a,b]\rightarrow \mathbb{R}$. We define an equivalence relation in $R$ as follows: $f$ and $g$ are said to be integrally equivalent iff they differ only on a (Lebesgue) measure zero set. Prove that the collection of these equivalence classes has a cardinality of a continuum.
Please let me remind that a function is Riemann integrable on $[a,b]$ iff it is bounded there and its discontinuity set there is a null set (has measure zero).
It is known that $R$ has cardinality of $2^{|\mathbb{R}|}$, that is, greater that the continuum. Also, the collection of null sets has the same cardinality.
Please give your opinions on how one could approach this problem. Complete proofs are also welcome. Thank you.