Effective versions of strong measure zero sets are developed for various levels of complexity and computability. It is shown that the sets can be equivalently defined using a generalization of supermartingales called odds supermartingales, success rates on supermartingales, predictors, and coverings. We show Borel's conjecture of a set having strong measure zero if and only if it is countable holds in the time and space bounded setting. At the level of computability this does not hold. We show the computable level contains sequences at arbitrary levels of the hyperarithmetical hierarchy by proving a correspondence principle yielding a condition for the sets of computable strong measure zero to agree with the classical sets of strong measure zero. An algorithmic version of strong measure zero using lower semicomputability is defined. We show that this notion is equivalent to the set of NCR reals studied by Reimann and Slaman, thereby giving new characterizations of this set. Effective strong packing dimension zero is investigated requiring success with respect to the limit inferior instead of the limit superior. It is proven that every sequence in the corresponding algorithmic class is decidable. At the level of computability, the sets coincide with a notion of weak countability that we define.

翻译：针对不同复杂度和可计算性层次，本文发展了强测度零集的有效版本。研究表明，这些集合可以通过称为赔率超鞅的超鞅推广、超鞅上的成功率、预测器以及覆盖等方式等价定义。我们证明了在时间和空间有界设定下，波雷尔猜想——即一个集合具有强测度零当且仅当它是可数的——成立。在可计算性层面上，该结论不成立。通过证明一个对应原理（该原理为计算强测度零集与经典强测度零集的一致性提供了条件），我们表明可计算层次包含超算术层次任意级别的序列，从而证实了这一点。本文还定义了基于下半可计算性的强测度零算法版本。我们证明该概念等价于Reimann和Slaman所研究的NCR实数集，从而为此集合提供了新的表征。此外，本文研究了有效强填充维零（要求相对于下极限而非上极限成功），并证明相应算法类中的每个序列都是可判定的。在可计算性层面上，这些集合与我们定义的弱可数性概念相吻合。