Is $( L_p[a,b], d_p )$ is a metric space?

Question

Is $(L_p[a,b], d_p)$ a metric space where $d_p(f,g) = (\int_{b}^{a} |f(x) - g(x)|^p dx)^\frac{1}{p}$ ? where $f \in L_p[a,b] \iff ||f||_p = (\int_{b}^{a} |f|^p dx)$ for $1 \leq p < \infty$

Answer 1

Since it is written that $L^{p}$ instead of $\mathcal{L}^{p}$, we have $[f],[g]\in L^{p}$ and $[f]=[g]$ if and only if $f(x)=g(x)$ a.e. by definition.

So if $d([f],[g])=0$, then $|f(x)-g(x)|=0$ a.e., and hence $f(x)=g(x)$ a.e. and so $[f]=[g]$.