402 
IMPROPER L-INTEGRALS 
Since/is limited, we may clioose the constant O, such that 
g = f(x) + Q > 0, in 21. 
Then g is ¿-integrable, and hence, by case I o , g is measurable. 
Hence/, differing only by a constant from g, is also measurable. 
2. Let 21 be measurable. If f is L-integrable in 21, it is measur 
able in 2Í, and conversely. 
This follows from 1 and 409, 1. 
3. From 2 and 409, 3, we have at once the theorem : 
When the field of integration is measurable, an L-integrable func 
tion is integrable in Lebesgue's sense, and conversely ; moreover, both 
have the same value. 
Remark. In the theory which has been developed in the fore 
going pages, the reader will note that neither the field of integra 
tion nor the integrand needs to be measurable. This is not so in 
Lebesgue’s theory. In removing this restriction, we have been 
able to develop a theory entirely analogous to Riemann’s theory of 
integration, and to extend this to a theory of upper and lower in 
tegration. We have thus a perfect counterpart of the theory 
developed in Chapter XIII of vol. I. 
4. Let 21 be metric or complete. If f (x^ • • • xf) is limited and 
R-integrable, it is a measurable function in 21. 
For by 381, 2, it is ¿-integrable. Also since 2Ï is metric or 
complete, 2Í is measurable. We now apply 1. 
IMPROPER L-INTEGRALS 
Upper and Loieer Integrals 
425. 1. We propose now to consider the case that the integrand 
/(/j ••• xf) is not limited in the limited field of integration 21- In 
chapter II we have treated this case for ¿-integrals. To extend 
the definitions and theorems there given to ¿-integrals, we have 
in general only to replace metric or complete sets by measurable 
sets; discrete sets by null sets; unmixed sets by separated sets ;