Exploring Royden's Real Analysis: Measure Theory vs. Riesz Approach

[Castilla]

Does someone has read Royden's Real Analysis?

If so, please tell me if he teachs Lebesgue integration by way of measure theory or by way of Riesz´s approach (upper functions).

Thanks.

 

[Hurkyl]

I don't know exactly what those approaches are, so I'll sketch his text.

He starts by defining Lesbegue measure.

He defines a simple function to be a (finite) linear combination of characteristic functions.

The Lesbegue measure allows us to define the integral of a simple function that vanishes outside a set of finite measure.

The integral of a bounded measurable function defined on a set with finite measure (which I'll call "type *") is taken to be the infimum of the integrals of all simple functions which are nowhere less than f.

Then, the integral of a nonnegative measurable function over a measurable set E is taken to be the supremum of all "type *" functions that are nowhere greater than f.

Then finally, the integral of an arbitrary function is given by splitting it up into the difference of nonnegative functions and subtracting the integrals.

 

[Castilla]

Thanks for answering and for the data, Hurkyl.

Riesz approach avoids measure theory in the presentation of the basics of lebesgue integral.

He define first the "lebesgue integral" for step functions; then for "upper functions"; then for "lebesgue" functions; then for "measurable functions". Just from here onwards he introduces concepts of measure theory.