**Łukasz D ˛ebowski**

Institute of Computer Science, Polish Academy of Sciences, ul. Jana Kazimierza 5, 01-248 Warszawa, Poland; ldebowsk@ipipan.waw.pl; Tel.: +48-22-3800-553

Received: 20 November 2019; Accepted: 8 January 2020; Published: 9 January 2020

**Abstract:** We supply corrected proofs of the invariance of completion and the chain rule for the Shannon information measures of arbitrary fields, as stated by D ˛ebowski in 2009. Our corrected proofs rest on a number of auxiliary approximation results for Shannon information measures, which may be of an independent interest. As also discussed briefly in this article, the generalized calculus of Shannon information measures for fields, including the invariance of completion and the chain rule, is useful in particular for studying the ergodic decomposition of stationary processes and its links with statistical modeling of natural language.

**Keywords:** Shannon information measures; fields; invariance of completion; chain rule

**MSC:** 94A17
