K-randomness and Martin-Lof randomness are among many formalizations of randomness of infinite sequences, and these two are known to be equivalent. We can naturally modify the former to the definition of partial randomness. However, it is not obvious how to modify the latter to the definition of partial randomness. In this paper, we show that we can modify Martin-Lof randomness to a definition of partial randomness that is equivalent to the definition obtained by naturally modifying K-randomness. The basic idea is to modify the notion of measures used in the definition of Martin-Lof tests.