The functional equation \((1)\quad \sum^{n}_{i=1}\sum^{m}_{j=1}f(p_ iq_ j)=\sum^{n}_{i=1}p_ i^{\alpha}\sum^{m}_{j=1}f(q_ j)+\sum^{m}_{j=1}q_ j^{\beta}\sum^{n}_{i=1}f(p_ i)\) and its many generalizations known as “sum form functional equations” have been studied by the reviewer, the author and many others. Recently the reviewer and the author [Analysis 9, 253-267 (1989; Zbl 0683.39007)] have obtained the general solution of \((2)\quad \sum^{n}_{i=1}\sum^{m}_{j=1}F(p_ iq_ j)=\sum^{n}_{i=1}M_ 1(p_ i)\sum^{m}_{j=1}F(q_ j)+\sum^{m}_{j=1}M_ 2(q_ j)\sum^{n}_{i=1}F(p_ i)\) for fixed m,n\(\geq 3\) where F, \(M_ i:]0,1[^ k\to {\mathbb{R}}\), \(M_ i\) are non- constant multiplicative functions which are not additive. In the present paper, the author obtains the measurable solutions of (2) where F, \(M_ i:[0,1]\to {\mathbb{R}}\) are measurable with \(M_ 2(0)=0=M_ 1(0)\), \(M_ 1(1)=1=M_ 2(1)\) for m,n\(\geq 3\).