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A "cousin" of a theorem of Csáki and Fischer

A 1963 theorem of P. Csáki and J. Fischer deals with the "maximal correlation coefficient" in the context of independent pairs of $σ$-fields on a probability space. Here a somewhat restricted "cousin" of their result is presented for the same context, but involving in part an analogous measure of dependence based only on correlations of indicator functions. It was first proved by the author in an unpublished 1978 Ph.D. thesis. An example is constructed to show a limitation of this "cousin". Also, this "cousin" is used to trivially embellish a very sharp 2013 example of R. Peyre in connection with the comparison of these two measures of dependence.