21(5 Mr Di^on, On a propei'ty of summable functions. 

 The same method enables us to prove that 



Lt f\f{a' + e)-f{xyilv = if f \f{x)\da; 



e-*.0 J a J a 



exists, by means of the inequality 



|/(.r + 0) -fiw) I < \f{w) I + \;\x + e)\. 



5. The deduction of the continuity of 



1 f{t + x)f{t)dt, 



J a 



and that of the Fourier expansion for it, still hold good, and in 

 particular it follows that the series ^a^ + S (rt,f + t,f) converges 

 to the sum 





even when f{d') is unlimited, if this last integral exists and is 

 finite. 



6. It follows from the results of § 2 that a necessary condition 

 for a limited function /\it') to be summable is that the superior* 

 integral of \f{'i'+(^)—f{ii^ tend to zero with 0. The question 

 is at ouce suggested whether this condition is suthcieut, and if not 

 whether a satisfactory detinition can be given for the integral of a 

 function which satisfies this condition, but is not assumed to be 

 summable. 



* In Lebesigne's sense, see Hobson, F. E. V. p. 577. 



