By Stokes theorem, $\int_{X}dd^{c}\varphi\wedge T\wedge\omega^{n-p-1}=0$, hence

$$||\omega_{\varphi}\wedge T||:=\int_{X}\omega_{\varphi}\wedge T\wedge\omega^{n-p-1}=\int_{X}T\wedge\omega^{n-p}:=||T||.$$

Consider now $\psi\in L^{1}(T)$. Since $T$ has measure coefficients, this simply means that $\psi$ is integrable with respect to the total variation of these measures. Assume first $\psi\leq 0$, $\varphi\geq 0$ and $\varphi,\psi$ are smooth. Then

$$||\psi||_{L^{1}(T\wedge\omega_{\varphi})}:=\int_{X}(-\psi)T\wedge\omega_{\varphi}\wedge\omega^{n-p-1}=||\psi||_{L^{1}(T)}+\int_{X}(-\psi)T\wedge dd^{c}\varphi\wedge\omega^{n-p-1}.$$

Now it follows from Stokes theorem that

$$\int_{X}(-\psi)T\wedge dd^{c}\varphi\wedge\omega^{n-p-1}=\int_{X}\varphi T\wedge(-dd^{c}\psi)\wedge\omega^{n-p-1}$$

$$\leq\int_{X}\varphi T\wedge\omega^{n-p}\leq\sup_{X}\varphi\int_{X}T\wedge\omega^{n-p},$$

where the forelast inequality follows from $\varphi T\wedge\omega^{n-p}\geq 0$ and $-dd^{c}\psi\leq\omega$. This yields

$$||\psi||_{L^{1}(T\wedge\omega_{\varphi})}\leq||\psi||_{L^{1}(T)}+\sup_{X}\varphi||T||.$$

The general case follows by regularizing $\varphi,\psi$, observing that $\omega_{\varphi}=\omega_{\varphi^{\prime}}$ where $\varphi^{\prime}=\varphi-\inf_{X}\varphi\geq 0$, and decomposing $\psi=\psi^{\prime}+\sup_{X}\psi$ with $\psi^{\prime}=\psi-\sup_{X}\psi\leq 0$. ∎