Prove $\mathcal{L}$ is a complex vector space

Question

We define $\mathcal{L}$ as the space of integrable functions in a vector space.

Formally, given $(\Omega, \Sigma, \mu)$ a measure space, we define $\mathcal{L}(\Omega, \Sigma, \mu) := \{f:\Omega \rightarrow \mathbb{C} | \int{|f| d \mu} < \infty\}$.

So we have to prove this is a vector space over complex numbers. I’ve only found proofs of $L_p$ spaces being Banach spaces, but that’s not what I need.

Answer 1

If you want to prove that something is a vector space, you have to check the axioms https://en.wikipedia.org/wiki/Vector_space#Definition

Note here that the elements of your space are functions $f$. Thus for example if you want to check if the addition of $f_1$ and $f_2$ are in $\mathcal{L}$ for $f_1,f_2\in\mathcal{L}$, you have to check the condition $$\int |f_1+f_2|d\mu<\infty$$

You do this by using the linearity of the integral and the fact, that $f_1,f_2\in\mathcal{L}$. Thus, in this example $$\int |f_1+f_2|d\mu \leq \underbrace{\int |f_1|d\mu}_{<\infty}+\underbrace{\int |f_2|d\mu}_{<\infty}<\infty$$

The first inequality here is the triangle inequality. Do a similar computation for the other axioms to show that $\mathcal{L}$ is a vector space.