Let $f: [a, b] \to \mathbb R$ limited and integrable function and $g: [a, b] \to \mathbb R$ function. Presume that there exists points $y_1, y_2, ... , y_k \in [a, b]$ s. t. $f(x) = g(x)$ for all $x \in [a, b] - y_1, y_2,...,y_k.$ Then $g$ is also integrable on $[a,b]$ and
$\int_a^b g(x) dx = \int_a^b f(x) dx$
I didnt really get a grasp of this, how do I prove it in a neat way?
Note that $h:[a,b]\to \Bbb R$ defined by $h(x)=f(x)-g(x)$ is integrable and has integral zero since it is not zero at at most finitely many points. Hence since $f$ is integrable, $h+f$=$g$ is also integrable, and has integral $$\int_{[a,b]} g = \int_{[a,b]} h+f = \int_{[a,b]} h + \int _{[a,b]}f = \int_{[a,b]} f$$