Lebesgue point
In mathematics, given a locally Lebesgue integrable function f {\displaystyle f} on R k {\displaystyle \mathbb {R} ^{k}} , a point x {\displaystyle x} in the domain of f {\displaystyle f} is a Lebesgue point if lim r → 0 + 1 λ ( B ( x , r ) ) ∫ B ( x , r ) | f ( y ) − f ( x ) | d y = 0. {\displaystyle \lim _{r\rightarrow 0^{+}}{\frac {1}{\lambda (B(x,r))}}\int _{B(x,r)}\! |f(y)-f(x)|\,\mathrm {d} y=0.} Here, B ( x , r ) {\displaystyle B(x,r)} is a ball centered at x {\displaystyle x} with radius r > 0 {\displaystyle r>0} , and λ ( B ( x , r ) ) {\displaystyle \lambda (B(x,r))} is its Lebesgue measure.