Firstly, let me say that my knowledge of measure theory is, at best, limited.
My question is, if you have four Lebesgue integrable functions that are pair-wise almost everywhere equal and you take the convolution of the "non-equal" functions, are resulting functions also almost everywhere equal?
In other words:
Let $f_1, f_2, g_1, g_2 : \mathbb{R} \to [0,1]$ be Lebesgue integrable functions such that $f_1 = g_1$ almost everywhere and $f_2 = g_2$ almost everywhere.
Does it hold that $(f_1 * f_2) = (g_1 * g_2)$ almost everywhere?
Any pointers on how to prove (or disprove) this will be helpfull and much appreciated.