See :

• Thomas Hawkins, Lebesgue's theory of integration, AMS (1975), page 122, regarding Lebesgue's thesis "Intégrale, longueur, aire," published in 1902:

$m(E)$ is a nonnegative measure on bounded sets $E$, such that:

(1) $m(E) \ne 0$ for some set $E$;

(2) $m(E + a) = m(E)$ for every real number $a$;

(3) if $E_n$ are pairwise disjoint, then $m(\bigcup E_n)=\sum m(E_N)$.

See also : G.T.Q. Hoare and N.J.Lord, "Intégrale, longueur, aire" the Centenary of the Lebesgue Integral, The Mathematical Gazette, Vol. 86 (2002).

See :

• Thomas Hawkins, Lebesgue's theory of integration, AMS (1975), page 122, regarding Lebesgue's thesis "Intégrale, longueur, aire," published in 1902:

$m(E)$ is a nonnegative measure on bounded sets $E$, such that:

(1) $m(E) \ne 0$ for some set $E$;

(2) $m(E + a) = m(E)$ for every real number $a$;

(3) if $E_n$ are pairwise disjoint, then $m(\bigcup E_n)=\sum m(E_N)$.

See also : G.T.Q. Hoare and N.J.Lord, "Intégrale, longueur, aire" the Centenary of the Lebesgue Integral, The Mathematical Gazette, Vol. 86 (2002).

And see also : David Bressoud, A radical approach to Lebesgue's theory of integration, Cambridge (2008), Ch.5 The Development of Measure Theory, on Borel and Lebesgue.

See :

• Thomas Hawkins, Lebesgue's theory of integration, AMS (1975), page 122, regarding Lebesgue's thesis "Intégrale, longueur, aire," published in 1902:

$m(E)$ is a nonnegative measure on bounded sets $E$, such that:

(1) $m(E) \ne 0$ for some set $E$;

(2) $m(E + a) = m(E)$ for every real number $a$;

(3) if $E_n$ are pairwise disjoint, then $m(\bigcup E_n)=\sum m(E_N)$.

See :

• Thomas Hawkins, Lebesgue's theory of integration, AMS (1975), page 122, regarding Lebesgue's thesis "Intégrale, longueur, aire," published in 1902:

$m(E)$ is a nonnegative measure on bounded sets $E$, such that:

(1) $m(E) \ne 0$ for some set $E$;

(2) $m(E + a) = m(E)$ for every real number $a$;

(3) if $E_n$ are pairwise disjoint, then $m(\bigcup E_n)=\sum m(E_N)$.

See also : G.T.Q. Hoare and N.J.Lord, "Intégrale, longueur, aire" the Centenary of the Lebesgue Integral, The Mathematical Gazette, Vol. 86 (2002).