Now that we have an idea of what the Lebesgue integral is, how do we define it?

If we limit ourselves to bounded, measurable functions on the real line, we could do the following:

suppose there is a real number such that over . Then for some integer we could set up the partition of the range:

Now set up

And then have and

Note that plays the role of the lower sum and plays the role of the upper sum. If the function is integrable (as it always is if is measurable and bounded) and we define

Note: It is possible to define the Lebesgue integral without having the concept of measurable function first: we can start with a bounded function

and partition the range of by We can then look at all partitions of by measurable sets.

Then consider the characteristic functions so we can form a type of general step function and and call integrable if . Think of the first function as approximating from below by generalized step functions, and the second as approximating from above.

Note: one of the big deals about the Lebesgue integral is that we get better convergence properties; that is, if we have a sequence of integrable

functions $f_{n}\rightarrow f$ pointwise over a measurable set, then with only mild extra hypothesis, we can show that the limit function is also

integrable and that the integral can be obtained as some sort of limit of integrals.

But to make any headway on such theorems, we’ll have to retreat to some theorems concerning measurable sets; so far we’ve shown that some sets are not measurable; we haven’t developed sufficient conditions for a set to be measurable.

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