CRP-CAP:

High level transcription of the lac operon requires the presence of a specific activator protein known as CAP (catabolite activator protein), also known as cAMP receptor protein (CRP). This protein which is a dimer that not bind to DNA unless it is complexed with 3 5 cyclic AMP (cAMP). The CRP-cAMP complex connects to the lac promoter just upstream from the binding site for RNA polymerase. It raise the binding of RNA polymerase and so stimulates transcription of the lac operon.

Whether or not the cAMP receptor protein protein is capable to bind to the lac promoter depends on the carbon source available to the bacterium that is shown in the figure. When glucose is present, E. coli does not required to use lactose as a carbon source and so the lac operon does not required to be active. Therefore the system has evolved to be responsive to glucose. The Glucose inhibits adenylate cyclase, the enzyme which synthesizes cAMP from ATP. Therefore,  in the presence  of glucose  the intracellular  stages of cAMP  falls so CRP cannot bind to the lac promoter and the lac operon is only weakly active even in the  presence of  lactose. When the glucose is absent adenylate cyclase is not inhibited; the stages of intracellular cAMP rise and bind to CRP. Thus, when glucose is absent but lactose is present the CRP-cAMP complex stimulates transcription of the lac operon and permits the lactose to be used as another carbon source. In the absence of lactose the lac repressor of course ensures in which the lac operon remains inactive.  These combined controls ensure which the lacZ, lacY and lacA genes are transcribed strongly only if glucose is lactose and absent is present.

Figure:  Inducer inactivates the lac repressor and so allows transcription of the structural genes.