Posted by J. Holmer on November 12, 2007, 3:56 pm
Board Administrator
The idea is to introduce a sequence of functions f_n(x,y) each of which is measurable such that f_n \to f pointwise. We then appeal to the fact that the limit of a sequence of measurable functions is measurable to conclude that f is measurable.
The sequence f_n is defined as follows. Partition the x axis into intervals of length 1/N. For each y, define f_n(x,y) for x in the interval j/N to (j+1)/N as the linear function (in x) connecting f(j/N,y) to f((j+1)/N,y).
Specifically, on j/N <= x <= (j+1)/N, define
f_n(x,y) = N(f( (j+1)/N, y) - f(j/N,y)) ( x - j/N) + f(j/N, y)
f_n thus defined is continuous in (x,y) (not merely separately continuous) and is thus measurable.
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