Theorem: Integral of Integrable Function is Homogeneous

Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f: X \to \overline \R$ be a $\mu$-integrable function.
Let $\lambda \in \R$.
Let $\lambda f$ be the pointwise $\lambda$-multiple of $f$.
Then $\lambda f$ is $\mu$-integrable, and:
:$\ds \int \lambda f \rd \mu = \lambda \int f \rd \mu$

Proof:
First suppose that $\lambda \ge 0$. 
From Positive Part of Multiple of Function, we have: 
:$\paren {\lambda f}^+ = \lambda f^+$
From Negative Part of Multiple of Function, we have:
:$\paren {\lambda f}^- = \lambda f^-$
From Function Measurable iff Positive and Negative Parts Measurable, we have: 
:$f^-$ and $f^+$ are $\Sigma$-measurable.
From Pointwise Scalar Multiple of Measurable Function is Measurable, we have: 
:$\lambda f^-$, $\lambda f^+$ and $\lambda f$ are $\Sigma$-measurable. 
Then, we have: 
{{begin-eqn}}
{{eqn | l = \int \paren {\lambda f}^+ \rd \mu
      | r = \int \lambda f^+ \rd \mu
}}
{{eqn | r = \lambda \int f^+ \rd \mu
      | c = Integral of Positive Measurable Function is Positive Homogeneous
}}
{{eqn | o = <
      | r = \infty
      | c = since $f$ is $\mu$-integrable
}}
{{end-eqn}}
Similarly: 
{{begin-eqn}}
{{eqn | l = \int \paren {\lambda f}^- \rd \mu
      | r = \int \lambda f^- \rd \mu
}}
{{eqn | r = \lambda \int f^- \rd \mu
      | c = Integral of Positive Measurable Function is Positive Homogeneous
}}
{{eqn | o = <
      | r = \infty
      | c = since $f$ is $\mu$-integrable
}}
{{end-eqn}}
So: 
:$\lambda f$ is $\mu$-integrable. 
We then have: 
{{begin-eqn}}
{{eqn | l = \int \lambda f \rd \mu
      | r = \int \paren {\lambda f}^+ \rd \mu - \int \paren {\lambda f}^- \rd \mu
      | c = {{Defof|Integral of Measure-Integrable Function}}
}}
{{eqn | r = \lambda \int f^+ \rd \mu - \lambda \int f^- \rd \mu
      | c = Positive Part of Multiple of Function, Negative Part of Multiple of Function
}}
{{eqn | r = \lambda \paren {\int f^+ \rd \mu - \int f^- \rd \mu}
}}
{{eqn | r = \lambda \int f \rd \mu
      | c = {{Defof|Integral of Measure-Integrable Function}}
}}
{{end-eqn}}
Now suppose that $\lambda < 0$. 
From Positive Part of Multiple of Function, we have: 
:$\paren {\lambda f}^+ = -\lambda f^-$
From Negative Part of Multiple of Function, we have: 
:$\paren {\lambda f}^- = -\lambda f^+$ 
From Function Measurable iff Positive and Negative Parts Measurable, we have: 
:$f^-$ and $f^+$ are $\Sigma$-measurable.
From Pointwise Scalar Multiple of Measurable Function is Measurable, we have: 
:$\lambda f^-$, $\lambda f^+$ and $\lambda f$ are $\Sigma$-measurable.
Then we have: 
{{begin-eqn}}
{{eqn | l = \int \paren {\lambda f}^+ \rd \mu
      | r = \int \paren {-\lambda f^-} \rd \mu
}}
{{eqn | r = -\lambda \int f^- \rd \mu
      | c = Integral of Positive Measurable Function is Positive Homogeneous, since $-\lambda \ge 0$
}}
{{eqn | o = <
      | r = \infty
      | c = since $f$ is $\mu$-integrable
}}
{{end-eqn}}
Similarly:
{{begin-eqn}}
{{eqn | l = \int \paren {\lambda f}^- \rd \mu
      | r = \int \paren {-\lambda f^+} \rd \mu
}}
{{eqn | r = -\lambda \int f^+ \rd \mu
      | c = Integral of Positive Measurable Function is Positive Homogeneous
}}
{{eqn | o = <
      | r = \infty
      | c = since $f$ is $\mu$-integrable
}}
{{end-eqn}}
So: 
:$\lambda f$ is $\mu$-integrable. 
We then have: 
{{begin-eqn}}
{{eqn | l = \int \lambda f \rd \mu
      | r = \int \paren {\lambda f}^+ \rd \mu - \int \paren {\lambda f}^- \rd \mu
      | c = {{Defof|Integral of Measure-Integrable Function}}
}}
{{eqn | r = \int \paren {-\lambda f^-} \rd \mu - \int \paren {-\lambda f^+} \rd \mu
      | c = Positive Part of Multiple of Function, Negative Part of Multiple of Function
}}
{{eqn | r = \lambda \int f^+ \rd \mu - \lambda \int f^- \rd \mu
      | c = Integral of Positive Measurable Function is Positive Homogeneous
}}
{{eqn | r = \lambda \paren {\int f^+ \rd \mu - \int f^- \rd \mu}
}}
{{eqn | r = \lambda \int f \rd \mu
      | c = {{Defof|Integral of Measure-Integrable Function}}
}}
{{end-eqn}}
{{qed}}