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Does anyone notice a definition problem in royden's book?

In chapter 4, section 3, royden's definition: the integral of a nonnegative measurable function f over a measurable set E to be the supremum of all the integrals of bounded measurable functions (Each of them vanishes outside a set of finite measure and is no greater than f) over E.

Here is the problem: assume the measure of E is infinite, then an integral of a bounded measurable function (vanishes outside a set of finite measure and no greater than f) over E is not defined, because in section 2 royden only defines an integral of a bounded measurable function over a set withfinite measure. So there seems to be a problem in royden's definition above. Strictly speaking, integrals of bounded measurable functions over E are not defined before taking supremum.

If you have a background of real analysis before reading royden's book, then this definition problem could be solved by other means. However, if you are a beginner and your knowledge of real analysis is limited to the materials from chapter 1 to chapter 4, section 2 in royden's book, then this definition problem seems to be unsolvable.

Hope I make my point clear, thanks for any help!

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# Definition problem of the integral of a nonnegative f in royden's book

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