I have one more question about the Lebesgue integral:(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Lebesgue integral once again

**Physics Forums | Science Articles, Homework Help, Discussion**