A function that is Lebesgue integrable but not measurable (not absurd obviously)

I think: A function $f$, as long as it is measurable, though Lebesgue integrable or not, always has Lebesgue integral on any domain $E$.

However Royden & Fitzpatrick’s book "Real Analysis" (4th ed) seems to say implicitly that “a function could be integrable without being Lebesgue measurable”. In particular, theorem 7 page 103 says:

“If function $f$ is bounded on set $E$ of finite measure, then $f$ is Lebesgue integrable over $E$ if and only if $f$ is measurable”.

The book spends a half page to prove the direction “$f$ is integrable implies $f$ is measurable”! Even the book “Real Analysis: Measure Theory, Integration, And Hilbert Spaces” of Elias M. Stein & Rami Shakarchi does the same job!

This makes me think there is possibly a function that is not bounded, not measurable but Lebesgue integrable on a set of infinite measure?

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Update: Read the answer of smnoren and me below about the motivation behind the approaches to define Lebesgue integrals.
Final conclusion: The starting statement above is still true and doesn't contradict with the approach of Royden and Stein.