Consider a function f continuous on a compact interval.
Approximate it by a sequence of polygonal functions (you can).
Then consider a sequence of primitives of the polygonal functions (you can).
At last consider the limit of the latter sequence (you can).
Now you have found a primitive of f (you know) without integration.
I doubt that such a thing was shown for the first time in 1905.
Lebesgue's good faith is beyond discussion of course.
Do you know something about ?