Using the covering game, we prove that every (lightface)-set of positive Lebesgue measure contains a member which is arithmetical in . This result generalizes a result for due to Sacks and Tanaka.
"A measure theoretic basis theorem for ." J. Math. Soc. Japan 52 (2) 335 - 341, April, 2000. https://doi.org/10.2969/jmsj/05220335