Please please please please please I want some help ,Is there and body here who can help me in this question :
Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$ , how can I prove that :
$1)$ $h$ is an integrable function
$2)$ $\int hdμ= \int fdμ- \int gdμ$
I'm presuming that by integrable you mean the integrals of the functions are finite, and by positive you mean that the functions only take positive values.
$h$ will not generally be defined in a strict sense ($\infty -\infty$ is undefined, and depending on which functions $f,g$ you are considering, that could happen), but it will be defined almost everywhere (the functions are integrable, and thus take $\infty$ on a set of measure $0$), which is all that's needed.
To show this, I'd simply show the more general linearity theorem,
$\int \alpha f + \beta g dμ= \alpha \int fdμ+ \beta \int gdμ$ for $\alpha ,\beta \in \mathbb R$
1, Show $\int f + g dμ= \int fdμ+ \int gdμ$ is true for simple (here, I mean non-negative) functions.
2, Show the above is true for any $f,g$ that are non negative (Levi).
3, Show that the above is true for any $f,g$ integrable (the respective positive/negative parts of $h=f+g$ are integrable functions, and $h^+ - h^- = f^+ - f^- + g^+ - g^-$ can be re-arranged into $ h^+ +f^- +g^- = h^- + f^+ + g^+$ on which you can use 2 above).
4, show that $\int -fdμ = -\int fdμ$.