Lowpass equivalent spectrum

I'm studing bandpass to lowpass conversion. My book says the following:

Lowpass equivalent spectrum, simply equals the positive-frequency portion of
band pass signal, translated down to the origin.

My question is about the signal energy, following this rule energy of
baseband signal is 1/2 energy of passband signal. In frequency the signal
has been truncated so:

int ( -inf , +inf, band_pass_signl(f) df ) = 1/2 int ( -inf , +inf,
low_pass_equivalent_signl(f) df )

Right?

Why we define Lowpass equivalent?

Thank you very much for your support!


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