Let $\mathcal{L}^1$ be the space of all Lebesgue integrable functions, that is $f\in \mathcal{L}^1$ is measurable and $\int f^+d\mu,\int f^-d\mu<\infty$.
Show that this is a vector space , I know all I need to show is that for $f,g\in \mathcal{L}^1$ and $a,b\in \mathbb R$:
$af+bg\in \mathcal{L}^1$
So:
$$\int (af+bg)d\mu=a\int fd\mu +b\int gd\mu=a\int f^+d\mu -a\int f^- d\mu+b\int g^+d\mu-b\int g^-d\mu=\int(af^++bg^+)d\mu-\int(af^-+bg^-)d\mu \leq\left|\int(af^++bg^+)d\mu+\int(af^-+bg^-)d\mu\right|$$
And here is where I am stuck.
How do I go on from here, If it is a good direction? If not what is?
By definition $ f\in \mathcal{L}^1$ if
$$\int |f|d\mu <\infty .$$
Let $f,g\in \mathcal{L}^1$ and $a,b\in \mathbb R$, thus $af+bg\in \mathcal{L}^1$ if $\displaystyle\int |af+bg|d\mu<\infty$, so $$\int |af+bg| d\mu < \int (|af|+|bg|)d\mu=\int|af|d\mu+\int|bg|d\mu=|a|\int|f|d\mu+|b|\int|g|d\mu <\infty,$$ since $f,g\in \mathcal{L}^1$.