Linearity of Integration

Question

Let $f$ and $g$ are bounded, measurable functions on set of finite measure $E$.

I am trying to show linearity of integration. I have showed for $\alpha \ge 0$, $\int_E \alpha f = \alpha \int_E g$

I have confusion in $\int_E (f+g) = \int_E f + \int_E g$ part.

Let $\psi_1$ and $\psi_2$ two simple functions such that $f \le \psi_1 $ and $g \le \psi_2$ (thus $f+g \le \ psi_1 + \psi_2)$

$$\int_E(f+g) \le \int_E (\psi_1 + \psi_2) = \int_E \psi_1 + \int_E \psi_2 $$

Since $\int_E(f+g)$ is any lower bound, it must be less than or equal to the greatest one namely infimum.

$\int_E(f+g) \le \inf(\int_E (\psi_1 + \psi_2)) = \inf( \int_E \psi_1 + \int_E \psi_2)$

But how can I say $\inf( \int_E \psi_1 + \int_E \psi_2) = \int_E f + \int_E g $ ??

Is $\inf( \int_E \psi_1 + \int_E \psi_2)$ equal to $inf( \int_E \psi_1) +inf(\int_E \psi_2)$ ??

Thanks for any help. I’m in trouble about it

Answer 1

It is true that $\inf A_1+A_2= \inf A_1+\inf A_2$, where $A_1,A_2\subset \mathbb R$ and $A_1+A_2:=\{a_1+a_2:a_1\in A_1,\,a_2\in A_2\}$? If so set $A_1=\{\int\psi_1:\phi_1\ge f\}$ and $A_2=\{\int\psi_2:\phi_2\ge g\}$ and use linearity for simple functions. (It may help to write out the sets your are taking the infimum over)