Dominated convergence theorem

Hello

I think I misunderstood Lebesgue's Dominated Convergence Theorem, I
would like some help.

According to the theorem, if {f_n} is a sequence of real valued,
u-measurable functions defined on a set X, if {f_n} converges almost
everywhere to a function f and if there's an integrable function
function g:X-->R such that |f_n| <= g for all n, then Integral (f du) =
lim (Integral (f_n du)), the integrals taken over X.

If all the functions f_n are non-negative, we can define g =
supremum{f_n}, and then g is real valued (doesn't take on oo),
non-negative and measurable, so that g is integrable. In addition, g
dominates each |f_n|, which implies the dominated convergence theorem
holds and therefore Integral (f du) = lim (Integral (f_n du)). So, I
came to the very general conclusion that this is true for every
sequence of u-measurable, non-negative real valued functions, provided
it converges in R to a function f.

If we combine supremums and infimums, we can conclude this holds for
every sequence of u-measurable, real valued functions, provided it
converges in R to a function f.

There's something wrong with my reasoning, but I don't see what it is.

Thank you
Artur

.

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