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I have one more question about the Lebesgue integral:

What if we defined the Lebesgue integral like this:

Let X be a measurable space and f any nonnegative function from X to R.

Then the Lebesgue integral of f as [tex]\int_X f d\mu = sup(I_X)[/tex] where [tex]I_X[/tex] is the integral of a simple function and the sup is taken over all simple measurable functions on X, such that 0<=s<=f.

As you see this definition is the same as the original, except, that the assumption that f is measurable is missing.

My question is: What would be wrong with this definition?

What if we defined the Lebesgue integral like this:

Let X be a measurable space and f any nonnegative function from X to R.

Then the Lebesgue integral of f as [tex]\int_X f d\mu = sup(I_X)[/tex] where [tex]I_X[/tex] is the integral of a simple function and the sup is taken over all simple measurable functions on X, such that 0<=s<=f.

As you see this definition is the same as the original, except, that the assumption that f is measurable is missing.

My question is: What would be wrong with this definition?

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