Accurate current measurements on a DSO are normally taken with a Current probe of which there are 2 main types: Current transformer (CT) and Hall Effect.

A combination of both allows both AC and DC measurements to be taken.

This is easily accomplished with a Current probe and DSO by changing the **Ch input units** from V to A(I) and then selecting the correct attenuation to match your current probe. eg. 10 mA/mV

Most scope owners are aware Current Probes are expensive items, often costing many times the value of entry level DSO's.

With complex Power analysis capabilities of some DSO's there are other measurement factors to consider: timing Skew between Voltage and Current probes.

This error is usually only nS or pS and does not affect basic measurement to any significant degree.

This arises from the propagation delays between the 2 probes as a result of differing cable lengths and risetime delays in Current probes. For accurate comparative measurements probes must be "De-Skewed" using the DSO processing to negate any time difference between probes.

Current measurements can also be taken using a shunt method and calculations should be made to determine the voltage amplitude as to how it converts to A/div.

Like any measurement techniques, all current measurements will affect the DUT and the resultant waveforms to some degree.

This ^.

The issues of propagation and phase delay are important if you are concerned with accurate power calculations. Look for "De-Skew" functionality in the scope's settings, or as Rigol calls it on the 1054z, "Delay-Cal" in the second page of the Channel settings menu. This is especially important if you are using some kind of inductive (coil) pickup for current sensing. Do you know the phase delay introduced by the coil? Can you synch it properly to your voltage probe, to give accurate power computations? Try doing a simultaneous measurement of current, by looking at voltage drop across a current-viewing resistor (little or no phase delay), and the coil pickup probe indication (unknown phase and propagation delay). Are they exactly synchronous, or is there some "skew" in the coil pickup that needs to be corrected to match the time constant of a voltage probe, as it must be for accurate instantaneous power computation?

Why are these current probes so expensive? What do they contain? Maybe their components aren't so expensive?

They contain _engineering_. The components themselves may be relatively cheap, but getting them to work together properly and accurately isn't so cheap.

Do the oscilloscope need to support I too? Do oscilloscopes from companies such as Rigol, Siglent and Hantek support it too?

All an oscilloscope is, is a time-domain voltmeter. But many DSOs can _display_ a voltage reading, such as the voltage drop across a current-viewing resistor, in units of Current (amps, mA, etc). Even the Rigol 1054Z can do this much... although the "amps" label doesn't carry through to the "measurements" displays, which can be a little confusing. The Rigol also has a ridiculously large number of "probe attenuation" values available. Using these values with the appropriate CVR one can essentially do the Ohm's Law calculation in the background so that the scope trace "current" values are accurate.

When I started to learn electronics, it always puzzled me that oscilloscopes measures voltage but not current. Why not? From my POV, measuring current in an oscilloscope form could help in many other ways too.

Since we know and understand Ohm's Law, we can easily convert a voltage drop through a known resistance, into a current value. With a multichannel scope one can use two channels and the math features to perform a differential voltage measurement of voltage drop across a current-viewing resistive element just about anywhere in a circuit, giving the current through that element. So the scope can measure current, with the correct use of current probes or current-viewing resistors.