1. Key concept: molar conductivity

   Molar conductivity is defined as $\Lambda_m = \frac{\kappa \times 1000}{C}$ where:

   • $\kappa$ = conductivity of solution
   • $C$ = concentration in mol L$^{-1}$

2. Behavior at infinite dilution

   The solution is stated to be an infinitely dilute aqueous solution of a weak electrolyte.

   For a weak electrolyte, dilution increases degree of ionization, so ordinarily $\Lambda_m$ increases on dilution.

   However, at infinite dilution, the molar conductivity has already approached its limiting value: $\Lambda_m \to \Lambda_m^\circ$

   Once this limiting value is reached, further dilution does not produce any meaningful increase in molar conductivity.

3. What happens when volume is doubled?

   Doubling the volume by adding water decreases concentration further, but since the solution is already infinitely dilute, the molar conductivity is effectively at its maximum limiting value.

   So theoretically, it should remain unchanged.

4. Practical issue in measurement

   At infinite dilution, conductivity $\kappa$ becomes extremely small because the number of ions per unit volume is very low. Hence experimental measurement becomes difficult and inaccurate.

   Therefore, the best statement is:

   • molar conductivity remains same, or
   • it cannot be measured accurately under such extreme dilution.

5. Check options

   • A: depend upon type of electrolyte — not appropriate here.
   • B: increase sharply — false; at infinite dilution it has already reached limiting value.
   • C: decrease sharply — false.
   • D: remain same or cannot be measured accurately — correct.

Final Answer

$\boxed{\text{D}}$