Let $(X,\mathcal S)$ be a measurable space. Let $f : X \longrightarrow \Bbb R^*$ be a map. Then the following are equivalent.
$(1)$ $f^{-1} (c, \infty] \in \mathcal S$ for every $c \in \Bbb R.$
$(2)$ $f^{-1} [c, \infty] \in \mathcal S$ for every $c \in \Bbb R.$
$(3)$ $f^{-1} [-\infty, c) \in \mathcal S$ for every $c \in \Bbb R.$
$(4)$ $f^{-1} [-\infty, c] \in \mathcal S$ for every $c \in \Bbb R.$
$(5)$ $f^{-1} \{\infty\}, f^{-1} \{-\infty\} \in \mathcal S$ and $f^{-1} (E) \in \mathcal S$ for every $E \in \mathcal B_{\Bbb R}.$
If $f$ will satisfy any of the properties $(1) - (5)$ mentioned above $f$ is said to be an $\mathcal S$-measurable function.
Now for any $f : X \longrightarrow \Bbb R^*$ define two functions $f^+$ and $f^{-}$ in the following way $:$
$$ f^+(x) = \left\{ \begin{array}{ll} f(x) & \quad \text {if}\ f(x) \geq 0 \\ 0 & \quad \text {if}\ f(x) \lt 0 \end{array} \right. $$
$$ f^-(x) = \left\{ \begin{array}{ll} -f(x) & \quad \text {if}\ f(x) \leq 0 \\ 0 & \quad \text {if}\ f(x) \gt 0 \end{array} \right. $$
Now I am studying these things from NPTEL lecture series on Measure and Integration by Inder K. Rana. What he claimed in one of his lectures is the following $:$
A function $f : X \longrightarrow \Bbb R^*$ is $\mathcal S$-measurable iff both $f^+$ and $f^-$ are $\mathcal S$-measurable. He claimed that $$ ({f^+})^{-1}[c,\infty] = \left\{ \begin{array}{ll} f^{-1}[c,\infty] & \quad \text {if}\ c \geq 0 \\ f^{-1} [0,\infty] & \quad \text {if}\ c \lt 0 \end{array} \right. $$
Similar things happen for $f^{-}.$ I think he (the instructor) made a mistake here. I think instead the following will be correct.
$$({f^+})^{-1}[c,\infty] = \left\{ \begin{array}{ll} f^{-1}[c,\infty] & \quad \text {if}\ c \gt 0 \\ X & \quad \text {if}\ c \leq 0 \end{array} \right.$$
Can anybody please verify it? Thanks in advance.
Source $:$ https://youtu.be/R9E7vqJDn3I?list=PLtKWB-wrvn4mbGE2XeUbnVw1cwAj4-f0C&t=290