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We consider two quantities that measure complexity of binary strings: KM(x) is defined as the negative logarithm of continuous a priori probability on the binary tree, and K(x) denotes prefix complexity of a binary string x. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence ω such that the sum of 2KM(x)−K(x) over all prefixes x of ω is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets.

The Sum 2 KM(x)−K(x) Over All Prefixes x of Some Binary Sequence Can be Infinite