Absolute Value inside an integral

Question

So I have that $|f(x) - h(x)| \le |f(x) - g(x)| + |g(x) - h(x)|$.

What I'm wondering is if this is the same as saying

$$\int_a^b |f(x) - h(x)|{\rm d}x \le \int_a^b |f(x) - g(x)|{\rm d}x + \int_a^b |g(x) - h(x)|{\rm d}x.$$

Is this valid?

Answer 1

Yes, it is triangle inequality evaluated in the integral from a to b.

It is posible thanks to the monotonic property of the integrals.

The only condition is that f,g,h must be integrable on [a,b], indeed continouos in (a,b) so the rest of them would be.

https://es.wikipedia.org/wiki/Desigualdad_triangular#Generalizaci%C3%B3n_de_la_desigualdad_triangular