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The European Commission has ended an investigation into price-fixing by major e-book publishers after they pledged to keep the market competitive.

The Commission said commitments by five publishers, including Apple, had been made legally binding.

They have said they will not restrict prices, restoring "normal competitive conditions" in the books industry, it added.

Apple and others are being sued by the US over the pricing of e-books.

In Europe, the publishers are Apple, Simon & Schuster, Harper Collins, Hachette of France and Verlagsgruppe Georg von Holtzbrinck. Apple sells books through its iBooks platform on the iPad and iPhone.

When the probe was launched in December 2011, the UK's Penguin was included. The Commission said it was in talks with the publisher on the possible commitments it could make which "would allow an early closure of proceedings".

"While each separate publisher and each retailer of e-books are free to choose the type of business relationship they prefer, any form of collusion to restrict or eliminate competition is simply unacceptable," EU Competition Commissioner Joaquin Almunia said.

"The commitments proposed... will restore normal competitive conditions in this new and fast-moving market, to the benefit of the buyers and readers of e-books," he added.

In the US, Apple and Hachette, HarperCollins, Macmillan, Simon and Schuster and Penguin are accused of colluding over the prices of e-books they sell.

Hachette, HarperCollins and Simon and Schuster settled but the case is proceeding against Apple, Macmillan and Penguin "for conspiring to end e-book retailers' freedom to compete on price", the Justice Department said.